Negative resistance

{{Short description|Property that an increasing voltage results in a decreasing current}} {{Good article}} {{Use American English|date = April 2019}} [[File:Fluorescent light strip 2 tube.JPG|thumb|upright=1.3|Fluorescent lamp, 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 arc flash and destroy itself.<ref name="Sinclair" /><ref name="Aluf" /> 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.<ref name="Sinclair" />]]

In electronics, '''negative resistance''' ('''NR''') is a property of some electrical circuits and devices in which an increase in voltage across the device's terminals results in a decrease in electric current through it.<ref name="Amos"/>

This is in contrast to an ordinary resistor, in which an increase in applied voltage causes a proportional increase in current in accordance with Ohm's law, resulting in a positive resistance.<ref name="Shanefield"/> Under certain conditions, negative resistance can increase the power of an electrical signal, amplifying it.<ref name="Aluf" /><ref name="Gottlieb"/><ref name="Kaplan"/>

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 <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 amplify,<ref name="Aluf" /><ref name="Suzuki"/> converting DC power applied to its terminals to AC output power to amplify an AC signal applied to the same terminals.<ref name="Carr"/><ref name="Iniewski" /> They are used in electronic oscillators and amplifiers,<ref name="Shahinpoor"/> particularly at microwave frequencies. Most microwave energy is produced with negative differential resistance devices.<ref name="Golio"/> They can also have hysteresis<ref name="Kumar2"/> and be bistable, and so are used in switching and memory circuits.<ref name="Beneking"/> 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.<ref name="Kaplan" /><ref name="Gilmore" />

[[File:Ganna diode 3A703B.jpg|thumb|A Gunn diode, a semiconductor device with negative differential resistance used in electronic oscillators to generate microwaves. <ref name="Sinclair">{{cite book | last = Sinclair | first = Ian Robertson | title = Sensors and transducers, 3rd Ed. | publisher = Newnes | date = 2001 | pages = 69–70 | url = https://books.google.com/books?id=s_WIb91uKK8C&q=%22gas+discharge%22+%22negative+resistance&pg=PA69 | isbn = 978-0-7506-4932-2}}</ref><ref name="Kularatna">{{cite book |last = Kularatna |first = Nihal |title = Power Electronics Design Handbook |publisher = Newnes |date = 1998 |pages = 232–233 |url = https://books.google.com/books?id=IBx801tIgjYC&q=%22negative+resistance&pg=PA233 |isbn = 978-0-7506-7073-9 |url-status = live |archive-url = https://web.archive.org/web/20171221182853/https://books.google.com/books?id=IBx801tIgjYC&pg=PA233&lpg=PA233&dq=%22negative+resistance |archive-date = 2017-12-21 }}</ref> <ref name="Amos">{{cite book | last1 = Amos | first1 = Stanley William | last2 = Amos | first2 = Roger S. | last3 = Dummer | first3 = Geoffrey William Arnold | title = Newnes Dictionary of Electronics, 4th Ed. | publisher = Newnes | date = 1999 | page = 211 | url = https://books.google.com/books?id=lROa-MpIrucC&q=%22negative+resistance&pg=PA211 | isbn = 978-0-7506-4331-3}}</ref><ref name="Graf">{{cite book |last = Graf |first = Rudolf F. |title = Modern Dictionary of Electronics, 7th Ed. |publisher = Newnes |date = 1999 |page = 499 |url = https://books.google.com/books?id=AYEKAQAAQBAJ&q=%22negative+resistance&pg=PA499 |isbn = 978-0-7506-9866-5 |url-status = live |archive-url = https://web.archive.org/web/20171221182851/https://books.google.com/books?id=AYEKAQAAQBAJ&pg=PA499&dq=%22negative+resistance |archive-date = 2017-12-21 }}</ref> <ref name="Shanefield">{{cite book | last = Shanefield | first = Daniel J. | title = Industrial Electronics for Engineers, Chemists, and Technicians | publisher = Elsevier | date = 2001 | pages = 18–19 | url = https://books.google.com/books?id=DUmwY0QJk28C&pg=PA19 | isbn = 978-0-8155-1467-1}}</ref> While a positive resistance consumes power from current passing through it, a negative resistance produces power.<ref name="Groszkowski">{{cite book |last1 = Groszkowski |first1 = Janusz |title = Frequency of Self-Oscillations |publisher = Pergamon Press - PWN (Panstwowe Wydawnictwo Naukowe) |date = 1964 |location = Warsaw |pages = 45–51 |url = https://books.google.com/books?id=H_ZFBQAAQBAJ&pg=PA45 |isbn = 978-1-4832-8030-1 |url-status = live |archive-url = https://web.archive.org/web/20160405074841/https://books.google.com/books?id=H_ZFBQAAQBAJ&pg=PA45 |archive-date = 2016-04-05 }}</ref> <ref name="Gottlieb">{{cite book |last = Gottlieb |first = Irving M. |title = Practical Oscillator Handbook |publisher = Elsevier |date = 1997 |pages = 75–76 |url = https://books.google.com/books?id=e_oZ69GAuxAC&q=%22negative+resistance&pg=PA75 |isbn = 978-0-08-053938-6 |url-status = live |archive-url = https://web.archive.org/web/20160515053022/https://books.google.com/books?id=e_oZ69GAuxAC |archive-date = 2016-05-15 }}</ref> <ref name="Kaplan">{{cite journal |first = Ross M. |last = Kaplan |title = Equivalent circuits for negative resistance devices |version = Technical Report No. RADC-TR-68-356 |publisher = Rome Air Development Center, US Air Force Systems Command |pages = 5–8 |date = December 1968 |url = http://www.dtic.mil/dtic/tr/fulltext/u2/846083.pdf |access-date = September 21, 2012 |archive-url = https://web.archive.org/web/20140819082258/http://www.dtic.mil/dtic/tr/fulltext/u2/846083.pdf |archive-date = August 19, 2014 }}</ref> <ref name="Suzuki">"''In semiconductor physics, it is known that if a two-terminal device shows negative differential resistance it can amplify.''" {{cite journal |last1 = Suzuki |first1 = Yoshishige |last2 = Kuboda |first2 = Hitoshi |title = Spin-torque diode effect and its application |journal = Journal of the Physical Society of Japan |volume = 77 |issue = 3 |article-number = 031002 |date = March 10, 2008 |url = http://jpsj.ipap.jp/link?JPSJ/77/031002/ |doi = 10.1143/JPSJ.77.031002 |access-date = June 13, 2013 |bibcode = 2008JPSJ...77c1002S |url-status = live |archive-url = https://web.archive.org/web/20171221182851/http://jpsj.ipap.jp/link?JPSJ%2F77%2F031002%2F |archive-date = December 21, 2017 |url-access= subscription }}</ref> <ref name="Carr">{{cite book |last = Carr |first = Joseph J. |title = Microwave & Wireless Communications Technology |publisher = Newnes |date = 1997 |location = USA |pages = 313–314 |url = https://books.google.com/books?id=1j1E541LKVoC&q=%22negative+differential+resistance%22+amplify&pg=PA314 |isbn = 978-0-7506-9707-1 |url-status = live |archive-url = https://web.archive.org/web/20170707111723/https://books.google.com/books?id=1j1E541LKVoC&pg=PA314&dq=%22negative+differential+resistance%22+amplify |archive-date = 2017-07-07 }}</ref> <ref name="Shahinpoor">{{cite book | last1 = Shahinpoor | first1 = Mohsen | last2 = Schneider | first2 =Hans-Jörg | title = Intelligent Materials | publisher = Royal Society of Chemistry | date = 2008 | location = London | page = 209 | url = https://books.google.com/books?id=Hmq4ctnA1KIC&pg=PA209 | isbn = 978-0-85404-335-4}}</ref> <ref name="Kumar2">{{cite journal |last = Kumar |first = Umesh |title = Design of an indigenized negative resistance characteristics curve tracer |journal = Active and Passive Elect. Components |volume = 23 |pages = 1–2 |publisher = Hindawi Publishing Corp. |date = April 2000 |url = http://downloads.hindawi.com/journals/apec/2000/969073.pdf |access-date = May 3, 2013 |url-status = live |archive-url = https://web.archive.org/web/20140819125752/http://downloads.hindawi.com/journals/apec/2000/969073.pdf |archive-date = August 19, 2014 }}</ref> <ref name="Beneking">{{cite book |last = Beneking |first = H. |title = High Speed Semiconductor Devices: Circuit aspects and fundamental behaviour |publisher = Springer |date = 1994 |pages = 114–117 |url = https://books.google.com/books?id=HdDXZRioqWkC&q=%22negative+resistance+(NR)%22+oneport&pg=PA115 |isbn = 978-0-412-56220-4 |url-status = live |archive-url = https://web.archive.org/web/20171221182851/https://books.google.com/books?id=HdDXZRioqWkC&pg=PA115&lpg=PA115&dq=%22negative+resistance+(NR)%22+oneport |archive-date = 2017-12-21 }}</ref> <ref name="Golio">{{cite book |last = Golio |first = Mike |title = The RF and Microwave Handbook |publisher = CRC Press |date = 2000 |pages = 5.91 |url = https://books.google.com/books?id=UIHMnx0k9oAC&pg=SA5-PA91 |isbn = 978-1-4200-3676-3 |url-status = live |archive-url = https://web.archive.org/web/20171221182851/https://books.google.com/books?id=UIHMnx0k9oAC&pg=SA5-PA91 |archive-date = 2017-12-21 }}</ref>]]

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

The resistance between two terminals of an electrical device or circuit is determined by its current–voltage (''I–V'') curve (characteristic curve), giving the current <math>i</math> through it for any given voltage <math>v</math> across it.<ref name="Herrick">{{cite book | last = Herrick | first = Robert J. | title = DC/AC Circuits and Electronics: Principles & Applications | publisher = Cengage Learning | date = 2003 | pages = 106, 110–111 | url = https://books.google.com/books?id=E_wKgWBu8rUC&q=%22static+resistance&pg=PA110 | isbn = 978-0-7668-2083-8}}</ref> 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.<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.

Negative resistance occurs in a few nonlinear (nonohmic) devices.<ref name="Haisch">{{cite web |last = Haisch |first = Bernhard |title = Nonlinear conduction |work = Online textbook Vol. 1: DC Circuits |publisher = All About Circuits website |date = 2013 |url = http://www.allaboutcircuits.com/vol_1/chpt_2/6.html |access-date = March 8, 2014 |url-status = live |archive-url = https://web.archive.org/web/20140320120241/http://www.allaboutcircuits.com/vol_1/chpt_2/6.html |archive-date = March 20, 2014 }}</ref> In a nonlinear component the ''I–V'' curve is not a straight line,<ref name="Shanefield" /><ref name="Simpson">{{cite book |last = Simpson |first = R. E. |title = Introductory Electronics for Scientists and Engineers, 2nd Ed. |publisher = Addison-Wesley |date = 1987 |location = US |pages = 4–5 |url = http://www.physics.oregonstate.edu/~tgiebult/COURSES/ph411/Reading/simp1a.pdf |isbn = 978-0-205-08377-0 |archive-url = https://web.archive.org/web/20140819130019/http://www.physics.oregonstate.edu/~tgiebult/COURSES/ph411/Reading/simp1a.pdf |archive-date = 2014-08-19 |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 |last = Aluf |first = Ofer |title = Optoisolation Circuits: Nonlinearity Applications in Engineering |publisher = World Scientific |date = 2012 |pages = 8–11 |url = https://books.google.com/books?id=DRui7sQTwRYC&pg=PA9 |isbn = 978-981-4317-00-9 |url-status = live |archive-url = https://web.archive.org/web/20171221182851/https://books.google.com/books?id=DRui7sQTwRYC&pg=PA9 |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 |last = Lesurf |first = Jim |title = Negative Resistance Oscillators |work = The Scots Guide to Electronics |publisher = School of Physics and Astronomy, Univ. of St. Andrews |date = 2006 |url = http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/RadCom/part5/page1.html |access-date = August 20, 2012 |url-status = live |archive-url = https://web.archive.org/web/20120716211956/http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/RadCom/part5/page1.html |archive-date = July 16, 2012 }}</ref><ref name="Kaiser">{{cite book | title = Electromagnetic Compatibility Handbook | last = Kaiser | first = Kenneth L. | publisher = CRC Press | year = 2004 | isbn = 978-0-8493-2087-3 | pages = 13–52 | 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 |last = Simin |first = Grigory |title = Lecture 08: Tunnel Diodes (Esaki diode) |work = ELCT 569: Semiconductor Electronic Devices |publisher = Prof. Grigory Simin, Univ. of South Carolina |date = 2011 |url = http://www.ee.sc.edu/personal/faculty/simin/ELCT563/08%20Tunnel%20Diodes.pdf |access-date = September 25, 2012 |archive-url = https://web.archive.org/web/20150923233956/http://www.ee.sc.edu/personal/faculty/simin/ELCT563/08%20Tunnel%20Diodes.pdf |archive-date = September 23, 2015 }}, pp. 18–19,</ref>

[[File:Quadrants of IV plane.svg|thumb|The quadrants of the ''I–V'' plane,<ref name="Chua2" /><ref name="Traylor">{{cite web |last = Traylor |first = Roger L. |title = Calculating Power Dissipation |work = Lecture Notes – ECE112:Circuit Theory |publisher = Dept. of Elect. and Computer Eng., Oregon State Univ. |date = 2008 |url = http://web.engr.oregonstate.edu/~traylor/ece112/lectures/calc_power_diss.pdf |access-date = 23 October 2012 |url-status = live |archive-url = https://web.archive.org/web/20060906041611/http://web.engr.oregonstate.edu/~traylor/ece112/lectures/calc_power_diss.pdf |archive-date = 6 September 2006 }}, [https://web.archive.org/web/20150726145426/http://inst.eecs.berkeley.edu/~ee100/fa08/lectures/EE100supplementary_notes_3.pdf archived]</ref> 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:<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 (chord) from the origin through the point on the ''I–V'' curve.<ref name="Shanefield" /> In a power source, like a battery or electric generator, positive current flows ''out'' of the positive voltage terminal,<ref name="Crisson" /> opposite to the direction of current in a resistor, so from the passive sign convention <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 | last1 = Kouřil | first1 = František | last2 = Vrba | first2 = Kamil | title = Non-linear and parametric circuits: principles, theory and applications | publisher = Ellis Horwood | date = 1988 | page = 38 | url = https://books.google.com/books?id=jftSAAAAMAAJ&q=%22active+resistor%22+%22negative+resistance%22 | isbn = 978-0-85312-606-5 }}</ref> However this term is never used in practice, because the term "resistance" is only applied to passive components.<ref name="Karady">"''...since [static] resistance is always positive...the resultant power [from Joule's law] must also always be positive. ...[this] means that the resistor always absorbs power.''" {{cite book | last1 = Karady | first1 = George G. | last2 = Holbert | first2 = Keith E. | title = Electrical Energy Conversion and Transport: An Interactive Computer-Based Approach, 2nd Ed. | publisher = John Wiley and Sons | date = 2013 | pages = 3.21 | url = https://books.google.com/books?id=VzBMPDiCr84C&q=%22resistance+is+always+positive%22absorbs+power%22&pg=SA3-PA21 | isbn = 978-1-118-49803-3}}</ref><ref name="Bakshi">"''Since the energy absorbed by a (static) resistance is always positive, resistances are passive devices.''" {{cite book |last = Bakshi |first = U.A. |author2 = V.U.Bakshi |title = Electrical And Electronics Engineering |publisher = Technical Publications |date = 2009 |pages = 1.12 |url = https://books.google.com/books?id=9zePYs9v6QsC&q=%22energy+absorbed+22always+positive&pg=SA1-PA12 |isbn = 978-81-8431-697-1 |url-status = live |archive-url = https://web.archive.org/web/20171221182851/https://books.google.com/books?id=9zePYs9v6QsC&pg=SA1-PA12&dq=%22energy+absorbed+22always+positive |archive-date = 2017-12-21 }}</ref><ref name="Glisson">{{cite book |last = Glisson |first = Tildon H. |title = Introduction to Circuit Analysis and Design |publisher = Springer |date = 2011 |location = USA |pages = 114–116 |url = https://books.google.com/books?id=7nNjaH9B0_0C&q=%22passive+sign+convention%22++power+%22negative+resistance%22&pg=PA116 |isbn = 978-90-481-9442-1 |url-status = live |archive-url = https://web.archive.org/web/20171208211033/https://books.google.com/books?id=7nNjaH9B0_0C&pg=PA116&lpg=PA116&dq=%22passive+sign+convention%22++power+%22negative+resistance%22 |archive-date = 2017-12-08 }}, see footnote p. 116</ref> Static resistance determines the power dissipation in a component.<ref name="Traylor" /><ref name="Bakshi" /> Passive devices, which consume electric power, have positive static resistance; while active devices, which produce electric power, do not.<ref name="Simin" /><ref name="Morecroft">{{cite book |last = Morecroft |first = John Harold |author2 = A. Pinto |author3 = Walter Andrew Curry |title = Principles of Radio Communication |publisher = John Wiley and Sons |date = 1921 |location = US |page = [https://archive.org/details/PrinciplesOfRadioCommunication/page/n128 112] |url = https://archive.org/details/PrinciplesOfRadioCommunication }}</ref><ref name="Baker">{{cite book | last = Baker | first = R. Jacob | title = CMOS: Circuit Design, Layout, and Simulation | publisher = John Wiley & Sons | date = 2011 | pages = 21.29 | url = https://books.google.com/books?id=rCxNKzuBIAwC&q=%22negative+resistance%22+battery&pg=SA21-PA29 | isbn = 978-1-118-03823-9}} In this source "negative resistance" refers to negative static resistance.</ref>

*'''Differential resistance''' (also called ''dynamic'',<ref name="Aluf" /><ref name="Kaiser" /> or ''incremental''<ref name="Shanefield" /> 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,<ref name="Gottlieb" /> the inverse slope 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" />

Negative resistance, like positive resistance, is measured in ohms.

Conductance is the reciprocal of resistance.<ref name="Herrick2">{{cite book |last1 = Herrick |first1 = Robert J. |title = DC/AC Circuits and Electronics: Principles & Applications |publisher = Cengage Learning |date = 2003 |page = 105 |url = https://books.google.com/books?id=E_wKgWBu8rUC&q=%22conductance&pg=PA105 |isbn = 978-0-7668-2083-8 |url-status = live |archive-url = https://web.archive.org/web/20160410221245/https://books.google.com/books?id=E_wKgWBu8rUC&pg=PA105&dq=%22conductance |archive-date = 2016-04-10 }}</ref><ref name="Ishii">{{cite book |last1 = Ishii |first1 = Thomas Koryu |title = Practical microwave electron devices |publisher = Academic Press |date = 1990 |page = 60 |url = https://books.google.com/books?id=pRtTAAAAMAAJ&q=%22static+conductance%22+%22differential+conductance%22&pg=PA60 |isbn = 978-0-12-374700-6 |url-status = live |archive-url = https://web.archive.org/web/20160408183239/https://books.google.com/books?id=pRtTAAAAMAAJ&pg=PA60&q=%22static+conductance%22+%22differential+conductance%22 |archive-date = 2016-04-08 }}</ref> It is measured in siemens (formerly ''mho'') which is the conductance of a resistor with a resistance of one ohm.<ref name="Herrick2" /> Each type of resistance defined above has a corresponding conductance<ref name="Ishii" /> *'''Static conductance''' <math display="block">G_\mathrm{static} = \frac{1}{R_\mathrm{static}} = \frac{i}{v}</math> *'''Differential conductance''' <math display="block">g_\mathrm{diff} = \frac{1}{r_\mathrm{diff}} = \frac{di}{dv}</math> 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 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</ref> while a positive resistance will have a positive conductance.<ref name="Kouřil" /><ref name="Ishii" />

{{multiple image | align = center | direction = horizontal | header = | image1 = Ohmic resistance.svg | 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 <math>R_\text{static} = r_\text{diff} = {v \over i} > 0 </math> | width1 = 190 | image2 = Negative differential resistance definition.svg | caption2 = Fig. 2: ''I–V'' curve with negative differential resistance ''(red 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 <math>r_\text{diff} = \frac {\Delta v}{\Delta i} = \frac {v_2 - v_1}{i_2 - i_1} </math> Since <math>\Delta v\;>\;0</math> and <math>\Delta i < 0</math>, at point '''''P''''' <math>r_\text{diff} < 0</math>. | width2 = 230 | image3 = Negative static resistance definition.svg | caption3 = Fig. 3: ''I–V'' curve of a power source.<ref name="Simin" /> 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''''', <math>v < 0</math> and <math>i > 0</math>, so <math>R_\text{static} = \frac{v}{i} < 0 </math> | width3 = 152 | image4 = Active negative resistance definition.svg | caption4 = Fig. 4: ''I–V'' curve of a negative linear<ref name="Groszkowski" /> or "active" resistance<ref name="Chua2" /><ref name="Pippard2">{{cite book |last = Pippard |first = A. B. |title = The Physics of Vibration |publisher = Cambridge University Press |date = 2007 |pages = 350, fig. 36; p. 351, fig. 37a; p. 352 fig. 38c; p. 327, fig. 14c |url = https://books.google.com/books?id=F8-9UNvsCBoC&q=%22negative-resistance&pg=PA350 |isbn = 978-0-521-03333-6 |url-status = live |archive-url = https://web.archive.org/web/20171221182853/https://books.google.com/books?id=F8-9UNvsCBoC&pg=PA350&dq=%22negative-resistance |archive-date = 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.</ref><ref name="Butler" /> ''(AR, red)''. 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 }}

==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.<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. || 140px |- | 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 |last = Chua |first = Leon |title = Linear and Non Linear Circuits |publisher = McGraw-Hill Education |date = 2000 |pages = 49–50 |url = http://inst.eecs.berkeley.edu/~ee100/fa08/lectures/EE100supplementary_notes_3.pdf |isbn = 978-0-07-116650-8 |archive-url = https://web.archive.org/web/20150726145426/http://inst.eecs.berkeley.edu/~ee100/fa08/lectures/EE100supplementary_notes_3.pdf |archive-date = 2015-07-26 }},</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 electric field '''''E''''' (decreasing potential).<ref name="Traylor" /> <math>P = vi\;>\;0</math> so the charges lose potential energy doing work 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. || 140px |- | 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. || 140px |- | 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" /> || 150px |}

==Types and terminology== {| style="background:#f5f5f5; float:right;" border="1" cellpadding="5" cellspacing="0" style="margin: 1em auto 1em auto;" |- ! style="background:#cfcfcf;" | ! style="background:#cfcfcf; text-align:center;" | '''''r''diff > 0''' Positive differential resistance ! style="background:#cfcfcf; text-align:center;" border="1" | '''''r''diff < 0''' Negative differential resistance |- ! scope="row" style="background:#dfdfdf; text-align:center;" | '''''R''static > 0''' Passive: Consumes net power | valign="top" | Positive resistances:{{bulleted list| Resistors| Ordinary diodes| Most passive components}} | Passive negative differential resistances:{{bulleted list| Tunnel diodes| Gunn diodes| Gas-discharge tubes}} |- ! scope="row" style="background:#dfdfdf; text-align:center;" | '''''R''static < 0''' Active: Produces net power | valign="top" | Power sources:{{bulleted list| Batteries| Generators| Transistors| Most active components}} | "Active resistors" Positive feedback amplifiers used in:{{bulleted list| Feedback oscillators| Negative impedance converters| Active filters}} |} {{Clear}} 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>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 tuned circuit to make an oscillator.<ref name="Ghadiri" /><ref name="Razavi" /><ref name="Solymar" /> They can also have hysteresis.<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 |last = Reich |first = Herbert J. |title = Principles of Electron Tubes |publisher = McGraw-Hill |date = 1941 |location = US |page = 215 |url = http://www.tubebooks.org/Books/reich_principles.pdf |url-status = live |archive-url = https://web.archive.org/web/20170402091020/http://www.tubebooks.org/Books/reich_principles.pdf |archive-date = 2017-04-02 }} on Peter Millet's [http://www.tubebooks.org Tubebooks] {{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 | first = Sheila |author2=Hermann Schumacher |author3=Anand Gopinath | title = High-Speed Electronics and Optoelectronics: Devices and Circuits | publisher = Cambridge Univ. Press | date = 2009 | page = 388 | url = https://books.google.com/books?id=LRhN5wiNuXAC&pg=PA388 | isbn = 978-0-521-86283-7}}</ref><ref name="Deliyannis">{{cite book |last = Deliyannis |first = T. |author2 = Yichuang Sun |author3 = J.K. Fidler |title = Continuous-Time Active Filter Design |publisher = CRC Press |date = 1998 |pages = 82–84 |url = https://books.google.com/books?id=C8z40DAIhmYC&q=%22negative+resistance&pg=PA82 |isbn = 978-0-8493-2573-1 |url-status = live |archive-url = https://web.archive.org/web/20171221182851/https://books.google.com/books?id=C8z40DAIhmYC&pg=PA82&lpg=PA83&dq=%22negative+resistance |archive-date = 2017-12-21 }}</ref>

thumb

*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.<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 gas-discharge tubes, tunnel diodes, and Gunn diodes.<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 '''Negative differential resistance''' section below. {{clear}} thumb *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,<ref name="Aluf" /><ref name="Crisson" /><ref name="Wilson">{{cite web |last = Wilson |first = Marcus |title = Negative Resistance |work = Sciblog 2010 Archive |publisher = Science Media Center |date = November 16, 2010 |url = http://sciblogs.co.nz/physics-stop/2010/11/16/negative-resistance/ |access-date = September 26, 2012 |url-status = live |archive-url = https://web.archive.org/web/20121004161234/http://sciblogs.co.nz/physics-stop/2010/11/16/negative-resistance/ |archive-date = October 4, 2012 }}, [https://web.archive.org/web/20150726153639/http://sci.waikato.ac.nz/physicsstop/2010/11/negative-resistance.shtml archived]</ref><ref name="HorowitzVideo">{{cite web |last = Horowitz |first = Paul |title = Negative Resistor – Physics 123 demonstration with Paul Horowitz |work = Video lecture, Physics 123, Harvard Univ. |publisher = YouTube |date = 2004 |url = https://www.youtube.com/watch?v=qKqrXcU2jGo |access-date = November 20, 2012 |url-status = live |archive-url = https://web.archive.org/web/20151217083158/https://www.youtube.com/watch?v=qKqrXcU2jGo |archive-date = December 17, 2015 }} 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 transistors and op-amps with positive feedback can have this type of negative resistance,<ref name="Ghadiri" /><ref name="Pippard3" /><ref name="Crisson">{{cite journal | last = Crisson | first = George | title = Negative Impedances and the Twin 21-Type Repeater | journal = Bell System Tech. J. | volume = 10 | issue = 3 | pages = 485–487 | date = July 1931 | url = https://archive.org/details/bstj10-3-485 | doi = 10.1002/j.1538-7305.1931.tb01288.x | bibcode = 1931BSTJ...10..485C | access-date = December 4, 2012}}</ref><ref name="Deliyannis" /> and are used in feedback oscillators and active filters.<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 circuit theory this is called an "active resistor".<ref name="Chua2" /><ref name="Kouřil" /><ref name="Popa">{{cite book | last = Popa | first = Cosmin Radu | title = Synthesis of Analog Structures for Computational Signal Processing | publisher = Springer | date = 2012 | page = 323 | doi =10.1007/978-1-4614-0403-3_7 | isbn = 978-1-4614-0403-3| chapter = Active Resistor Circuits }}</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 positive feedback or ''regeneration''. These are covered in the '''Active resistors''' section below. {{clear}}

[[File:Battery IV curve showing negative static resistance.svg|thumb|upright=0.7|A battery 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="Simpson" /><ref name="Morecroft" /><ref name="Baker" /><ref name="Fett">{{cite journal | last = Fett | first = G. H. | title = Negative Resistance as a Machine Parameter | journal = Journal of Applied Physics | volume = 14 | issue = 12 | pages = 674–678 | date = October 4, 1943 | url = http://jap.aip.org/resource/1/japiau/v14/i12/p674_s1?isAuthorized=no | archive-url = https://archive.today/20140317074058/http://jap.aip.org/resource/1/japiau/v14/i12/p674_s1?isAuthorized=no | archive-date = March 17, 2014 | doi = 10.1063/1.1714945 | access-date = December 2, 2012 | bibcode = 1943JAP....14..674F | url-access = subscription }}, 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 '''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).<ref name="Babin">{{cite web |last = Babin |first = Perry |title = Output Impedance |work = Basic Car Audio Electronics website |date = 1998 |url = http://www.bcae1.com/outptimp.htm |access-date = December 28, 2014 |url-status = live |archive-url = https://web.archive.org/web/20150417095028/http://www.bcae1.com/outptimp.htm |archive-date = April 17, 2015 }}</ref><ref name="Glisson2">[https://books.google.com/books?id=7nNjaH9B0_0C&dq=%22output+resistance%22+battery+generator+amplifier&pg=PA96 Glisson, 2011 ''Introduction to Circuit Analysis and Design'', p. 96] {{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.

==List of negative resistance devices== Electronic components with negative differential resistance include these devices: *tunnel diode,<ref name="Fogiel">{{cite book | last = Fogiel | first = Max | title = The electronics problem solver | publisher = Research & Education Assoc. | date = 1988 | pages = 1032.B–1032.D | url = https://books.google.com/books?id=Zpwtq_SjKSoC&q=%22negative+resistance&pg=RA1-PA1029 | isbn = 978-0-87891-543-9}}</ref><ref name="Rybin">{{cite book | last = Rybin | first = Yu. K. | title = Electronic Devices for Analog Signal Processing | publisher = Springer | date = 2011 | pages = 155–156 | url = https://books.google.com/books?id=FdLRdC8epOEC&q=%22negative+resistance&pg=PA155 | isbn = 978-94-007-2204-0}}</ref> resonant tunneling diode<ref name="Iezekiel">{{cite book | last = Iezekiel | first = Stavros | title = Microwave Photonics: Devices and Applications | publisher = John Wiley and Sons | date = 2008 | page = 120 | url = https://books.google.com/books?id=3NIy4Qv6PCcC&q=%22negative+resistance%22+%22resonant+tunnelling+diode&pg=PA120 | isbn = 978-0-470-74486-4}}</ref> and other semiconductor diodes using the tunneling mechanism<ref name="Kapoor">{{cite book | last = Kapoor | first = Virender |author2=S. Tatke | title = Telecom Today: Application and Management of Information Technology | publisher = Allied Publishers | date = 1999 | pages = 144–145 | url = https://books.google.com/books?id=DA9fVnb8QbMC&pg=PA144 | isbn = 978-81-7023-960-4}}</ref> *Gunn diode<ref name="Radmanesh">{{cite book | last = Radmanesh | first = Matthew M. | title = Advanced RF & Microwave Circuit Design | publisher = AuthorHouse | date = 2009 | pages = 479–480 | url = https://books.google.com/books?id=YC6NFiFkJkQC&q=%22negative+resistance%22+gunn+impatt+tunnel&pg=PA479 | isbn = 978-1-4259-7243-1}}</ref> and other diodes using the transferred electron mechanism<ref name="Kapoor" /> *IMPATT diode,<ref name="Rybin" /><ref name="Radmanesh" /> TRAPATT diode and other diodes using the impact ionization mechanism<ref name="Kapoor" /> *Some NPN transistors with E-C reverse biased, known as negistor<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 |archive-url=https://web.archive.org/web/20060906055849/http://www.keelynet.com/zpe/negistor.htm |archive-date=2006-09-06 }}</ref> *unijunction transistor (UJT)<ref name="Fogiel" /><ref name="Rybin" /> *thyristors<ref name="Fogiel" /><ref name="Rybin" /> *triode and tetrode vacuum tubes operating in the dynatron mode<ref name="Gottlieb" /><ref name="Whitaker">{{cite book |last = Whitaker |first = Jerry C. |title = The electronics handbook, 2nd Ed. |publisher = CRC Press |date = 2005 |page = 379 |url = https://books.google.com/books?id=FdSQSAC3_EwC&q=triode+tetrode+%22negative+resistance&pg=PA379 |isbn = 978-0-8493-1889-4 |url-status = live |archive-url = https://web.archive.org/web/20170331220534/https://books.google.com/books?id=FdSQSAC3_EwC |archive-date = 2017-03-31 }}</ref> *Some magnetron tubes and other microwave vacuum tubes<ref name="Gilmour">{{cite book |last = Gilmour |first = A. S. |title = Klystrons, Traveling Wave Tubes, Magnetrons, Cross-Field Amplifiers, and Gyrotrons |publisher = Artech House |date = 2011 |pages = 489–491 |url = https://books.google.com/books?id=l_1egQKKWe4C&q=magnetron+%22negative+resistance&pg=PA490 |isbn = 978-1-60807-184-5 |url-status = live |archive-url = https://web.archive.org/web/20140728142143/http://books.google.com/books?id=l_1egQKKWe4C&pg=PA490&dq=magnetron+%22negative+resistance |archive-date = 2014-07-28 }}</ref> *maser<ref name="Illingworth">{{cite book | last = Illingworth | first = Valerie | title = Astronomy | publisher = Infobase Publishing | date = 2009 | page = 290 | url = https://books.google.com/books?id=_c-ZRNuooYoC&q=maser+%22negative+resistance&pg=PA290 | isbn = 978-1-4381-0932-9}}</ref> *parametric amplifier<ref name="Rao">{{cite book | last = Rao | first = R. S. | title = Microwave Engineering | publisher = PHI Learning Pvt. Ltd | date = 2012 | page = 440 | url = https://books.google.com/books?id=ZecSEXlJE0YC&q=maser+%22negative+resistance&pg=PA440 | isbn = 978-81-203-4514-0}}</ref>

Electric discharges through gases also exhibit negative differential resistance,<ref name="Raju">{{cite book |last1 = Raju |first1 = Gorur Govinda |title = Gaseous Electronics: Theory and Practice |publisher = CRC Press |date = 2005 |page = 453 |url = https://books.google.com/books?id=I7Qi5vb2nB4C&q=%22negative+resistance%22+%22glow+discharge%22&pg=PA453 |isbn = 978-0-203-02526-0 |url-status = live |archive-url = https://web.archive.org/web/20150322102031/https://books.google.com/books?id=I7Qi5vb2nB4C&pg=PA453&dq=%22negative+resistance%22+%22glow+discharge%22 |archive-date = 2015-03-22 }}</ref><ref name="Siegman">{{cite book |last1 = Siegman |first1 = A. E. |title = Lasers |publisher = University Science Books |date = 1986 |pages = [https://archive.org/details/lasers0000sieg/page/63 63] |url = https://archive.org/details/lasers0000sieg |url-access = registration |quote = neon negative resistance glow discharge. |isbn = 978-0-935702-11-8 }}, fig. 1.54</ref> including these devices

*electric arc<ref name="Ayrton" /> *thyratron tubes<ref name="Satyam">{{cite book |last = Satyam |first = M. |author2 = K. Ramkumar |title = Foundations of Electronic Devices |publisher = New Age International |date = 1990 |page = 501 |url = https://books.google.com/books?id=EIavtzVDG-IC&q=%22negative+resistance%22+thyratron&pg=PA501 |isbn = 978-81-224-0294-0 |url-status = live |archive-url = https://web.archive.org/web/20140910033602/http://books.google.com/books?id=EIavtzVDG-IC |archive-date = 2014-09-10 }}</ref> *neon lamp<ref name="Shanefield" /> *fluorescent lamp<ref name="Kularatna" /> *other gas discharge tubes<ref name="Sinclair" /><ref name="Rybin" />

In addition, active circuits with negative differential resistance can also be built with amplifying devices like transistors and op amps, using feedback.<ref name="Rybin" /><ref name="Ghadiri">{{cite thesis |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 |date = Fall 2011 |pages = 9–10 |url = http://era.library.ualberta.ca/public/datastream/get/uuid:a590efa3-a428-4823-88e3-f071bac3f1d0/DS1 |access-date = March 21, 2014 |url-status = live |archive-url = https://web.archive.org/web/20120628225402/https://era.library.ualberta.ca/public/datastream/get/uuid:a590efa3-a428-4823-88e3-f071bac3f1d0/DS1 |archive-date = June 28, 2012 |doi = 10.7939/R3N88J }}</ref><ref name="Pippard3">see "Negative resistance by means of feedback" section, {{cite book |last = Pippard |first = A. B. |title = The Physics of Vibration |publisher = Cambridge University Press |date = 2007 |pages = 314–326 |url = https://books.google.com/books?id=F8-9UNvsCBoC&q=%22negative-resistance&pg=PA350 |isbn = 978-0-521-03333-6 |url-status = live |archive-url = https://web.archive.org/web/20171221182853/https://books.google.com/books?id=F8-9UNvsCBoC&pg=PA350&dq=%22negative-resistance |archive-date = 2017-12-21 }}</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 current–voltage curve.<ref name="Kaplan" /><ref name="Rybin" />

== Negative static or "absolute" resistance == {{multiple image | align = right | direction = horizontal | image1 = Thermodynamics and positive resistance.svg | image2 = Thermodynamics and negative resistance.svg | 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 second law of thermodynamics<ref name="Solymar" /><ref name="Wilson" /><ref name="Thompson">{{cite journal |last = Thompson |first = Sylvanus P. |title = On the properties of a body having a negative electric resistance |journal = The Electrician |volume = 37 |issue = 10 |pages = 316–318 |publisher = Benn Bros. |location = London |date = July 3, 1896 |url = https://books.google.com/books?id=8vIfAQAAMAAJ&q=negative+resistance&pg=PA316 |access-date = June 7, 2014 |url-status = live |archive-url = https://web.archive.org/web/20171106041304/https://books.google.com/books?id=8vIfAQAAMAAJ&pg=PA316&lpg=PA316&dq=negative+resistance |archive-date = November 6, 2017 }} also see editorial, "Positive evidence and negative resistance", p. 312</ref><ref name="Grant" /><ref name="Cole">{{cite news |last = Cole |first = K.C. |title = Experts Scoff at Claim of Electricity Flowing With 'Negative Resistance' |newspaper = Los Angeles Times |location = Los Angeles |date = July 10, 1998 |url = https://www.latimes.com/archives/la-xpm-1998-jul-10-mn-2507-story.html |access-date = December 8, 2012 |url-status = live |archive-url = https://web.archive.org/web/20150808013836/http://articles.latimes.com/1998/jul/10/news/mn-2507 |archive-date = August 8, 2015 }} on [https://latimes.com Los Angeles Times website] {{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 | last = Klein | first = Sanford | author2 = Gregory Nellis | title = Thermodynamics | publisher = Cambridge University Press | date = 2011 | page = 206 | url = https://books.google.com/books?id=FR2gVgjj3EEC&pg=PA206 | isbn = 978-1-139-49818-0 }}</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>R_\text{static} = v / i</math>) can be negative.<ref name="Thompson" /><ref name="PhysicsForums">{{cite web |last = resonant.freq |title = Confusion regarding negative resistance circuits |work = Electrical Engineering forum |publisher = Physics Forums, Arizona State Univ. |date = November 2, 2011 |url = http://www.physicsforums.com/showthread.php?t=546744 |access-date = August 17, 2014 |url-status = live |archive-url = https://web.archive.org/web/20140819090715/http://www.physicsforums.com/showthread.php?t=546744 |archive-date = August 19, 2014 }}</ref> In electronics, the term "resistance" is customarily applied only to passive materials and components<ref name="Bakshi" /> – such as wires, resistors and diodes. These cannot have <math>R_\text{static} < 0</math> as shown by Joule's law {{nowrap|<math>P = i^2 R_\text{static}</math>.}}<ref name="Karady" /> A passive device consumes electric power, so from the passive sign convention <math>P \ge 0</math>. Therefore, from Joule's law {{nowrap|<math>R_\text{static} \geq 0</math>.}}<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 |last = Gibilisco |first = Stan |title = Physics Demystified |publisher = McGraw Hill Professional |date = 2002 |page = 391 |isbn = 978-0-07-141212-4 }}</ref> For a passive device to have <math>R_\text{static} = v/i\;<\;0</math> would violate either conservation of energy<ref name="Aluf" /> or the second law of thermodynamics,<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 |last = Grant |first = Paul M. |title = Journey Down the Path of Least Resistance |work = OutPost on the Endless Frontier blog |publisher = EPRI News, Electric Power Research Institute |date = July 17, 1998 |url = http://www.w2agz.com/Publications/Opinion%20&%20Commentary/EPRI/OutPost/outpost4.pdf |access-date = December 8, 2012 |url-status = live |archive-url = https://web.archive.org/web/20130421103501/http://www.w2agz.com/Publications/Opinion%20%26%20Commentary/EPRI/OutPost/outpost4.pdf |archive-date = April 21, 2013 }} on [http://www.w2agz.com/ Paul Grant personal website] {{webarchive|url=https://web.archive.org/web/20130722172704/http://w2agz.com/ |date=2013-07-22 }}</ref> state that static resistance can never be negative.

[[File:Battery and resistor circuit 2.svg|thumb|upright=0.7|From KVL, the static resistance of a power source (''R''S), such as a battery, is always equal to the negative of the static resistance of its load (''R''L).<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 (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.<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 passive sign convention so the current and voltage have opposite signs, and their ratio is negative <math display="block">R_\mathrm{static} = \frac {v}{i} < 0 </math> This can also be proved from Joule's law<ref name="Simin" /><ref name="Morecroft" /><ref name="Thompson"/> <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>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 conservation of energy it is always simply equal to the negative of the static resistance of the attached circuit ''(right)''.<ref name="Morecroft" /><ref name="Deliyannis" />

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.<ref name="Aluf" /><ref name="Simin" /><ref name="Solymar">{{cite book | last = Solymar | first = Laszlo | author2 = Donald Walsh | title = Electrical Properties of Materials, 8th Ed. | publisher = Oxford University Press | date = 2009 | location = UK | pages = 181–182 | url = https://books.google.com/books?id=AiWyp0NQW6UC&q=%22negative+resistance%22+%22second+law%22+thermodynamics&pg=PA181 | isbn = 978-0-19-956591-7 }}</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 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.<ref name="Kaplan" /> Any active circuit or device with a finite power source is "''eventually passive''".<ref name="Miano">{{cite book |last = Miano |first = Giovanni |author2 = Antonio Maffucci |title = Transmission Lines and Lumped Circuits |publisher = Academic Press |date = 2001 |pages = 396, 397 |url = https://books.google.com/books?id=7McEEUwEHwgC&pg=PA396 |isbn = 978-0-12-189710-9 |url-status = live |archive-url = https://web.archive.org/web/20171009234344/https://books.google.com/books?id=7McEEUwEHwgC&pg=PA396&dq= |archive-date = 2017-10-09 }} This source calls negative differential resistances "passive resistors" and negative static resistances "active resistors".</ref><ref name="Chen">{{cite book |last = Chen |first = Wai-Kai |title = Nonlinear and distributed circuits |publisher = CRC Press |date = 2006 |pages = 1.18–1.19 |url = https://books.google.com/books?id=4VrQ2pITeSEC&q=%22eventually+passive&pg=SA1-PA18 |isbn = 978-0-8493-7276-6 |url-status = live |archive-url = https://web.archive.org/web/20170824224353/https://books.google.com/books?id=4VrQ2pITeSEC&pg=SA1-PA18&dq=%22eventually+passive |archive-date = 2017-08-24 }}</ref><ref name="Chua">see {{cite journal |last = Chua |first = Leon O. |title = Dynamic Nonlinear Networks: State of the Art |journal = IEEE Transactions on Circuits and Systems |volume = CAS-27 |issue = 11 |pages = 1076–1077 |publisher = Inst. of Electrical and Electronic Engineers |location = US |date = November 1980 |url = http://www.elettrotecnica.unina.it/files/demagistris/didattica/TdC/Chua_Dynamic_Circuits.pdf |access-date = September 17, 2012 |url-status = live |archive-url = https://web.archive.org/web/20140819102859/http://www.elettrotecnica.unina.it/files/demagistris/didattica/TdC/Chua_Dynamic_Circuits.pdf |archive-date = August 19, 2014 }} 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 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.

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 | last = Muthuswamy | author2 = Joerg Mossbrucker | title = A framework for teaching nonlinear op-amp circuits to junior undergraduate electrical engineering students | book-title = 2010 Conference Proceedings | publisher = American Society for Engineering Education | date = 2010 | url = http://search.asee.org/search/fetch;jsessionid=13wo7tplbh5np?url=file%3A%2F%2Flocalhost%2FE%3A%2Fsearch%2Fconference%2F32%2FAC%25202010Full182.pdf&index=conference_papers&space=129746797203605791716676178&type=application%2Fpdf&charset= | access-date = October 18, 2012 }}{{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''s will cause the amplifier to saturate, also making its resistance positive.

== Negative differential resistance == In a device or circuit with negative differential resistance (NDR), in some part of the ''I–V'' curve the current decreases as the voltage increases:<ref name="Lesurf" /> <math display="block">r_\mathrm{diff} = \frac {dv}{di} < 0 </math> The ''I–V'' curve is nonmonotonic (having peaks and troughs) with regions of negative slope representing negative differential resistance. {{multiple image | align = right | direction = vertical | header = Negative differential resistance | image1 = Voltage controlled negative resistance.svg | caption1 = Voltage controlled (N type) | width1 = 174 | image2 = Current controlled negative resistance.svg | caption2 = Current controlled (S type) | width2 = 170 | footer = }} Passive 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=== Negative differential resistances can be classified into two types:<ref name="Beneking" /><ref name="Kumar">{{cite book | last = Kumar | first = Anand | title = Pulse and Digital Circuits | publisher = PHI Learning Pvt. Ltd | date = 2004 | pages = 274, 283–289 | url = https://books.google.com/books?id=kwnF8YotcpUC&pg=PT289 | isbn = 978-81-203-2596-8}}</ref> *'''Voltage controlled negative resistance''' ('''VCNR''', ''short-circuit stable'',<ref name="Kumar" /><ref name="Tellegen">{{cite journal | last = Tellegen | first = B. d. h. | title = Stability of negative resistances | journal = International Journal of Electronics | volume = 32 | issue = 6 | pages = 681–686 | date = April 1972 | doi = 10.1080/00207217208938331 }}</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 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.</ref> 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.<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''1), then decreases in the region of negative resistance to a minimum (''i''2), then increases again. Devices with this type of negative resistance include the tunnel diode,<ref name="Fogiel" /> resonant tunneling diode,<ref name="Kidner">{{cite conference |first = C. |last = Kidner |author2 = I. Mehdi |author3 = J. R. East |author4 = J. I. Haddad |title = Potential and limitations of resonant tunneling diodes |book-title = First International Symposium on Space Terahertz Technology, March 5–6, 1990, Univ. of Michigan |page = 85 |publisher = US National Radio Astronomy Observatory |date = March 1990 |location = Ann Arbor, M |url = http://www.nrao.edu/meetings/isstt/papers/1990/1990084103.pdf |access-date = October 17, 2012 |url-status = live |archive-url = https://web.archive.org/web/20140819125435/http://www.nrao.edu/meetings/isstt/papers/1990/1990084103.pdf |archive-date = August 19, 2014 }}</ref> lambda diode, Gunn diode,<ref name="Du">{{cite book |last = Du |first = Ke-Lin |author2 = M. N. S. Swamy |title = Wireless Communication Systems: From RF Subsystems to 4G Enabling Technologies |publisher = Cambridge Univ. Press |date = 2010 |page = 438 |url = https://books.google.com/books?id=5dGjKLawsTkC&q=%22negative+resistance&pg=PA438 |isbn = 978-0-521-11403-5 |url-status = live |archive-url = https://web.archive.org/web/20171031123919/https://books.google.com/books?id=5dGjKLawsTkC&pg=PA438&dq=%22negative+resistance |archive-date = 2017-10-31 }}</ref> and dynatron oscillators.<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 IMPATT diode,<ref name="Du" /> UJT,<ref name="Fogiel" /> SCRs and other thyristors,<ref name="Fogiel" /> electric arc, and gas discharge tubes .<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 |last = Franz |first = Roger L. |title = Use nonlinear devices as linchpins to next-generation design |journal = Electronic Design Magazine |publisher = Penton Media Inc. |date = June 24, 2010 |url = http://electronicdesign.com/archive/use-nonlinear-devices-linchpins-next-generation-design |access-date = September 17, 2012 |url-status = live |archive-url = https://web.archive.org/web/20150618024847/http://electronicdesign.com/archive/use-nonlinear-devices-linchpins-next-generation-design |archive-date = June 18, 2015 }}, . An expanded version of this article with graphs and an extensive list of new negative resistance devices appears in {{cite web |last = Franz |first = Roger L. |title = Overview of Nonlinear Devices and Circuit Applications |work = Sustainable Technology |publisher = Roger L. Franz personal website |date = 2012 |url = http://home.comcast.net/~rgrhmmr/site/?/page/Nonlinear_Devices/&PHPSESSID=efc9c75dad7261ecb043447b8d58b7fd |access-date = September 17, 2012 }}</ref><ref name="Abraham">{{cite conference | first = George | last = Abraham | title = Multistable semiconductor devices and integrated circuits | book-title = Advances in Electronics and Electron Physics, Vol. 34–35 | pages = 270–398 | publisher = Academic Press | date = 1974 | url = https://books.google.com/books?id=mSX8-zZjgTsC&q=negative+resistance&pg=PA332 | isbn = 978-0-08-057699-2 | access-date = September 17, 2012}}</ref> These can have more than two stable states, and are of interest for use in digital circuits to implement multivalued logic.<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)'': <math display="block">\text{PVR} = i_1 / i_2 </math> The larger this is, the larger the potential AC output for a given DC bias current, and therefore the greater the efficiency

===Amplification=== {{multiple image | align = right | direction = horizontal | image1 = Tunnel diode amplifier.svg | width1 = 172 | image2 = Tunnel diode amplifier graph.svg | width2 = 150 | 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 load line <math>R</math> ''(blue)''.<ref name="Weaver" /> A small increase in input voltage, <math>v_i</math> ''(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 <math>v_o</math>. }}

A negative differential resistance device can amplify an AC signal applied to it<ref name="Suzuki" /><ref name="Shahinpoor" /> if the signal is biased 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 | first = Krzysztof | title = Wireless Technologies: Circuits, Systems, and Devices | publisher = CRC Press | date = 2007 | page = 488 | url = https://books.google.com/books?id=JJXrpazX9FkC&q=%22negative+resistance%22+amplification+bias&pg=PA488 | isbn = 978-0-8493-7996-3}}</ref>

The tunnel diode circuit ''(see diagram)'' is an example.<ref name="Weaver">{{cite web |last = Weaver |first = Robert |title = Negative Resistance Devices: Graphical Analysis and Load Lines |work = Bob's Electron Bunker |publisher = Robert Weaver personal website |date = 2009 |url = http://electronbunker.ca/eb/NegativeResistance.html |access-date = December 4, 2012 |url-status = live |archive-url = https://web.archive.org/web/20130204114156/http://electronbunker.ca/eb/NegativeResistance.html |archive-date = February 4, 2013 }}</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 voltage divider, 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 voltage gain 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=== {{multiple image | align = right | direction = horizontal | header = | 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. | width1 = 180 | image2 = Negative resistance AC equivalent circuit.svg | caption2 = AC equivalent circuit of NDR attached to external circuit.<ref name="Lowry">{{cite book |last = Lowry |first = H. R. |author2 = J. Georgis |author3 = E. Gottlieb |title = General Electric Tunnel Diode Manual, 1st Ed. |publisher = General Electric Corp. |date = 1961 |location = New York |pages = 18–19 |url = http://w140.com/Ge1961TunnelDiodeManual.pdf |url-status = live |archive-url = https://web.archive.org/web/20130512225930/http://w140.com/Ge1961TunnelDiodeManual.pdf |archive-date = 2013-05-12 }}</ref> The NDR acts as a dependent AC current source 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''S through the load ''R'', increasing the output power.<ref name="Lowry" /> | 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 superposition principle 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>)}}. <math display="block">v(t) = V_\text{bias} + \Delta v(t)</math> <math display="block">i(t) = I_\text{bias} + \Delta i(t)</math> 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° out of phase.<ref name="Carr" /><ref name="Radmanesh" /><ref name="Butler">{{cite web |last = Butler |first = Lloyd |title = Negative Resistance Revisited |work = Amateur Radio magazine |publisher = Wireless Institute of Australia, Bayswater, Victoria |date = November 1995 |url = http://users.tpg.com.au/users/ldbutler/NegativeResistance.htm |access-date = September 22, 2012 |url-status = live |archive-url = https://web.archive.org/web/20120914181641/http://users.tpg.com.au/users/ldbutler/NegativeResistance.htm |archive-date = September 14, 2012 }} on [http://users.tpg.com.au/users/ldbutler/index.htm Lloyd Butler's personal website] {{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 Heinrich Barkhausen in 1907 in [https://archive.org/details/dasproblemdersc01barkgoog ''Das Problem Der Schwingungserzeugung''] according to {{cite journal | last = Duncan | first = R. D. | title = Stability conditions in vacuum tube circuits | journal = Physical Review | volume = 17 | issue = 3 | page = 304 | date = March 1921 | url = https://books.google.com/books?id=rCgKAAAAIAAJ&q=%22negative+resistance&pg=PA304 | doi = 10.1103/physrev.17.302 | 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 ''[which]'' requires that the phase difference between voltage and current lie between 90° and 270°...''[and for nonreactive circuits]'' 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 equivalent circuit ''(right)'', the instantaneous AC current Δ''i'' flows through the device in the direction of ''increasing'' AC potential Δ''v'', as it would in a generator.<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 | first = Brian | title = Microwave Oscillators | work = Class Notes: ELEC 483 – Microwave and RF Circuits and Systems | publisher = Dept. of Elec. and Computer Eng., Queen's Univ., Ontario | date = 2006 | url = http://bmf.ece.queensu.ca/elec483/slides/oscillators.pdf | access-date = September 22, 2012 | pages = 4–9 }}{{Dead link|date=April 2020 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> <math display="block">P_\text{AC} = \Delta v \Delta i = r_\text{diff}|\Delta i|^2 < 0 </math> With the proper external circuit, the device can increase the AC signal power delivered to a load, serving as an amplifier,<ref name="Butler" /> 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.<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" /> <math display="block">|P_\text{AC}| \le I_\text{bias} V_\text{bias} </math> 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 reactance 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 | last = Chang | first = Kai | title = RF and Microwave Wireless Systems | publisher = John Wiley & Sons | date = 2000 | location = USA | pages = 139–140 | url = https://books.google.com/books?id=h7qVxk-AD-cC&q=impedance+real+imaginary+%22negative+resistance&pg=PA139 | isbn = 978-0-471-35199-3}}</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 |last = Maas |first = Stephen A. |title = Nonlinear Microwave and RF Circuits, 2nd Ed. |publisher = Artech House |date = 2003 |pages = 542–544 |url = https://books.google.com/books?id=SSw6gWLG-d4C&pg=PA542 |isbn = 978-1-58053-484-0 |url-status = live |archive-url = https://web.archive.org/web/20170225004613/https://books.google.com/books?id=SSw6gWLG-d4C&pg=PA542 |archive-date = 2017-02-25 }}</ref>

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 |last = Mazda |first = F. F. |title = Discrete Electronic Components |publisher = CUP Archive |date = 1981 |page = 8 |url = https://books.google.com/books?id=3qk8AAAAIAAJ&pg=PA9 |isbn = 978-0-521-23470-2 |url-status = live |archive-url = https://web.archive.org/web/20170803004744/https://books.google.com/books?id=3qk8AAAAIAAJ&pg=PA9&lpg=PA9&dq= |archive-date = 2017-08-03 }}</ref> <math display="block">P_{AC(rms)} \le \frac{1}{8}(v_2 - v_1)(i_1 - i_2) </math>

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

The reason that the output signal can leave a negative resistance through the same port that the input signal enters is that from transmission line theory, the AC voltage or current at the terminals of a component can be divided into two oppositely moving waves, the ''incident wave'' <math>V_I</math>, which travels toward the device, and the ''reflected wave'' <math>V_R</math>, which travels away from the device.<ref name="Bowick">{{cite book | last = Bowick | first = Chris Bowick | author2=John Blyler |author3=Cheryl J. Ajluni | title = RF Circuit Design, 2nd Ed. | publisher = Newnes | date = 2008 | location = USA | page = 111 | url = https://books.google.com/books?id=zpTnMsiUkmwC&pg=PA111 | isbn = 978-0-7506-8518-4}}</ref> A negative differential resistance in a circuit can amplify if the magnitude of its reflection coefficient <math>\Gamma </math>, the ratio of the reflected wave to the incident wave, is greater than one.<ref name="Gilmore">{{cite book | last1 = Gilmore | first1 = Rowan | last2 = Besser | first2 = Les |author-link2=Les Besser | title = Active Circuits and Systems | publisher = Artech House | date = 2003 | location = USA | pages = 27–29 | url = https://books.google.com/books?id=B_KGlNHSb9kC&pg=PA28 | isbn = 978-1-58053-522-9}}</ref><ref name="Frank" /> <math display="block">|\Gamma| \equiv \left|\frac{V_R}{V_I}\right| > 1 </math> where <math display="block">\Gamma \equiv \frac {Z_N - Z_L}{Z_N + Z_L} </math> 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 Smith chart, 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" />

===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.<ref name="Chen2">{{cite book |last = Chen |first = Wai Kai |title = The Electrical Engineering Handbook |publisher = Academic Press |date = 2004 |pages = 80–81 |url = https://books.google.com/books?id=qhHsSlazGrQC&q=nonlinear+%22multiple+operating+points&pg=PA80 |isbn = 978-0-08-047748-0 |url-status = live |archive-url = https://web.archive.org/web/20160819081609/https://books.google.com/books?id=qhHsSlazGrQC |archive-date = 2016-08-19 }}</ref> 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.<ref name="Dorf">{{cite book | last1 = Dorf | first1 = Richard C. | title = The Electrical Engineering Handbook | publisher = CRC Press | edition = 2 | date = 1997 | page = 179 | url = https://books.google.com/books?id=qP7HvuakLgEC&q=poles+stable+unstable&pg=PA179 | isbn = 978-1-4200-4976-3 }}</ref><ref name="Vukic">{{cite book |last1 = Vukic |first1 = Zoran |title = Nonlinear Control Systems |publisher = CRC Press |date = 2003 |pages = 53–54 |url = https://books.google.com/books?id=7SE6VAjyifgC&q=stability+unstable&pg=PA54 |isbn = 978-0-203-91265-2 |url-status = live |archive-url = https://web.archive.org/web/20171011065813/https://books.google.com/books?id=7SE6VAjyifgC&pg=PA54&dq=stability+unstable |archive-date = 2017-10-11 }}</ref> In contrast, a linear circuit has a single equilibrium point that may be stable or unstable.<ref name="Ballard">{{cite book | last1 = Ballard | first1 = Dana H. | title = An Introduction to Natural Computation | publisher = MIT Press | date = 1999 | page = 143 | url = https://books.google.com/books?id=WGYT2d4hgGAC&q=%22linear+system+has+a+single+equilibrium+point%22&pg=PA143 | isbn = 978-0-262-52258-8 }}</ref><ref name=Vukic1>[https://books.google.com/books?id=7SE6VAjyifgC&dq=%22one+equilibrium+state%22&pg=PA50 Vukic, Zoran (2003) ''Nonlinear Control Systems'', p. 50, 54]</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. 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) ''[https://books.google.com/books?id=UIHMnx0k9oAC&dq=golio+%22open+circuit+stable&pg=SA7-PA29 The RF and Microwave Handbook]'', pp. 7.25–7.26, 7.29</ref><ref name="Crisson2">Crisson (1931) ''[http://www3.alcatel-lucent.com/bstj/vol10-1931/articles/bstj10-3-485.pdf Negative Impedances and the Twin 21-Type Repeater] {{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 |first = M. A. |last = Karp |title = A transistor D-C negative immittance converter |version = APL/JHU CF-2524 |publisher = Advanced Physics Lab, Johns Hopkins Univ. |date = May 1956 |pages = 3, 25–27 |url = http://www.dtic.mil/dtic/tr/fulltext/u2/657144.pdf |access-date = December 3, 2012 |archive-url = https://web.archive.org/web/20140819125516/http://www.dtic.mil/dtic/tr/fulltext/u2/657144.pdf |archive-date = August 19, 2014 }} on US [http://www.dtic.mil/dtic/ Defense Technical Information Center] {{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 | last1 = Giannini | first1 = Franco | last2 = Leuzzi | first2 = Giorgio | title = Non-linear Microwave Circuit Design | publisher = John Wiley and Sons | date = 2004 | pages = 230–233 | url = https://books.google.com/books?id=GbZH_ApMKFwC&q=%22negative+resistance%22+%22negative+conductance&pg=PA233 | isbn = 978-0-470-84701-5 }}</ref> :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 | last1 = Yngvesson | first1 = Sigfrid | title = Microwave Semiconductor Devices | publisher = Springer Science & Business Media | date = 1991 | page = 143 | url = https://books.google.com/books?id=NYeIiO-91ukC&q=negative+resistance+stability&pg=PA143 | isbn = 978-0-7923-9156-2 }}</ref> <math display="block">Z_L + Z_N = R_L + R_N = R_L - r > 0 </math> so the CCNR is stable for<ref name="Beneking" /><ref name="Kumar" /><ref name="Crisson2" /> {{Equation box 1 |indent =:|cellpadding = 0 |border = 1 |border colour = black |background colour = transparent |equation = <math>R_L\;>\;r.</math> }} :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 conductance 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 conductance 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" /> {{Equation box 1 |indent =: |cellpadding = 0 |border = 1 |border colour = black |background colour = transparent |equation = <math>R_L < r.</math> }} :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 | first = J. T. | title = The Transistor as a Network Element | journal = Bell System Tech. J. | volume = 33 | issue = 2 | page = 330 |date=March 1954 | url = https://archive.org/details/bstj33-2-329 | doi = 10.1002/j.1538-7305.1954.tb03734.x | access-date = June 20, 2014| bibcode = 1954ITED....1....7B | s2cid = 51671649 }}</ref><ref group=note name="Confusion" />

For general negative resistance circuits with reactance, the stability must be determined by standard tests like the Nyquist stability criterion.<ref name="Gilmore2">{{cite book | last1 = Gilmore | first1 = Rowan | last2 = Besser | first2 = Les | author2-link = Les Besser | title = Practical RF Circuit Design for Modern Wireless Systems | publisher = Artech House | volume = 2 | date = 2003 | pages = 209–214 | url = https://books.google.com/books?id=B_KGlNHSb9kC&q=%22negative+resistance%22+%22nyquist+stability%22&pg=PA209 | isbn = 978-1-58053-674-5 }}</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 Smith chart.<ref name="Gilmore" />

===Operating regions and applications=== 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 load lines on the ''I–V'' curve<ref name="Kumar" /> ''(see graphs)''.

{{multiple image | align = right | direction = horizontal | image1 = Negative resistance stability regions VCNR.svg | caption1 = VCNR (N type) load lines and stability regions | image2 = Negative resistance stability regions CCNR.svg | caption2 = CCNR (S type) load lines and stability regions | width = 180 }}

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) (Q points) 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 voltage source. *CCNRs require a high impedance bias {{nowrap|(<math>R\;>\;r</math>)}} such as a current source, or voltage source in series with a high resistance. The AC load line (''L''1 − ''L''3) 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" /> *'''Stable region (green)''' (illustrated by line ''L''1): When the load line lies in this region, it intersects the ''I–V'' curve at one point ''Q''1.<ref name="Kumar" /> 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.<ref name="Gottlieb2">[https://books.google.com/books?id=e_oZ69GAuxAC&dq=%22negative+resistance&pg=PA106 Gottlieb 1997 ''Practical Oscillator Handbook'', pp. 105–108] {{webarchive|url=https://web.archive.org/web/20160515053022/https://books.google.com/books?id=e_oZ69GAuxAC |date=2016-05-15 }}</ref> **VCNRs are stable when <math>R_L < r</math>. **CCNRs are stable when <math>R_L > r</math>. *'''Unstable point''' (Line ''L''2): When <math>R_L = r</math> the load line is tangent to the ''I–V'' curve. The total differential (AC) resistance of the circuit is zero (poles on the ''jω'' axis), so it is unstable and with a tuned circuit can oscillate. Linear oscillators operate at this point. Practical oscillators actually start in the unstable region below, with poles in the RHP, but as the amplitude increases the oscillations become nonlinear, and due to ''eventual passivity'' the negative resistance ''r'' decreases with increasing amplitude, so the oscillations stabilize at an amplitude where<ref name="Nahin" /> <math>r = R_L</math>. *'''Bistable region (red)''' (illustrated by line ''L''3): In this region the load line intersects the ''I–V'' curve at three points.<ref name="Kumar" /> The center point (''Q''1) is a point of unstable equilibrium (poles in the RHP), while the two outer points, ''Q''2 and ''Q''3 are stable equilibria. So with correct biasing the circuit can be bistable, it will converge to one of the two points ''Q''2 or ''Q''3 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 are bistable when <math>R_L > r</math> **CCNRs are bistable when <math>R_L < r</math>

== Active resistors – negative resistance from feedback == {{multiple image | align = right | direction = horizontal | header = | image1 = Active negative resistance - voltage controlled.svg | width1 = 150 | image2 = Active negative resistance - current controlled.svg | width2 = 150 | image3 = Negative resistance vs loop gain.svg | width3 = 150 | footer = Typical ''I–V'' curves of "active" negative resistances:<ref name="Pippard2" /><ref name="Spangenberg">{{cite book |last1 = Spangenberg |first1 = Karl R. |title = Vacuum Tubes |publisher = McGraw-Hill |date = 1948 |page = 721 |url = http://www.tubebooks.org/Books/Spangenberg_vacuum_tubes.pdf |url-status = live |archive-url = https://web.archive.org/web/20170320032730/http://www.tubebooks.org/Books/Spangenberg_vacuum_tubes.pdf |archive-date = 2017-03-20 }}, fig. 20.20</ref> N-type ''(left)'', and S-type ''(center)'', generated by feedback amplifiers. These have negative differential resistance ''(red region)'' and produce power ''(grey region)''. Applying a large enough voltage or current of either polarity to the port moves the device into its nonlinear region where saturation of the amplifier causes the differential resistance to become positive ''('''black''' portion of curve)'', and above the supply voltage rails <math>\pm V_S</math> the static resistance becomes positive and the device consumes power. The negative resistance depends on the loop gain <math>A\beta </math> ''(right)''. }} thumb|upright=1.2|An example of an amplifier with positive feedback that has negative resistance at its input. The input current '''''i''''' is <math>i = \frac{v - Av}{R_1} + \frac{v}{R_\text{in}}</math> so the input resistance is <math>R = \frac{v}{i} = \frac{R_1}{1 + R_1/R_\text{in} - A}.</math> If <math>A > 1 + R_1/R_\text{in} </math> it will have negative input resistance.

In addition to the passive devices with intrinsic negative differential resistance above, circuits with amplifying devices like transistors or op amps can have negative resistance at their ports.<ref name="Aluf" /><ref name="Ghadiri" /> The input or output impedance of an amplifier with enough positive feedback applied to it can be negative.<ref name="Pippard3" /><ref name="Razavi">{{cite book | last = Razavi | first = Behzad | title = Design of Analog CMOS Integrated Circuits | publisher = The McGraw-Hill Companies | date = 2001 | pages = 505–506 | url = https://books.google.com/books?id=hl6JZ8DKlFwC&q=%22feedback+negative+resistance&pg=PA506 | isbn = 978-7-302-10886-3}}</ref><ref name="Armstrong">{{cite journal | last = Armstrong | first = Edwin H. | title = Some recent developments of regenerative circuits | journal = Proceedings of the IRE | volume = 10 | issue = 4 | pages = 244–245 | date = August 1922 | url = https://books.google.com/books?id=bNI1AQAAMAAJ&pg=PA244 | doi = 10.1109/jrproc.1922.219822 | bibcode = 1922PIRE...10..244A | s2cid = 51637458 | access-date = September 9, 2013}}. "Regeneration" means "positive feedback"</ref><ref name="SSBmanual">{{cite book | title = Technical Manual no. 11-685: Fundamentals of Single-Sideband Communication | publisher = US Dept. of the Army and Dept. of the Navy | date = 1961 | page = 93 | url = https://books.google.com/books?id=mcEXAAAAYAAJ&q=%22input+impedance+%22negative+resistance&pg=PA93 }}</ref> If <math>R_i</math> is the input resistance of the amplifier without feedback, <math>A</math> is the amplifier gain, and <math>\beta(j\omega)</math> is the transfer function of the feedback path, the input resistance with positive shunt feedback is<ref name="Aluf" /><ref name="Singh">{{cite book | last1 = Singh | first1 = Balwinder | last2 = Dixit | first2 = Ashish | title = Analog Electronics | publisher = Firewall Media | date = 2007 | page = 143 | url = https://books.google.com/books?id=gW24HmL2PrcC&q=%22input+resistance%22+%22output+resistance%22+feedback&pg=PA143 | isbn = 978-81-318-0245-8}}</ref> <math display="block">R_\text{if} = \frac {R_\text{i} }{1 - A\beta} </math> So if the loop gain <math>A\beta </math> is greater than one, <math>R_{if}</math> will be negative. The circuit acts like a "negative linear resistor"<ref name="Aluf" /><ref name="HorowitzVideo" /><ref name="Dimopoulos">{{cite book |last = Dimopoulos |first = Hercules G. |title = Analog Electronic Filters: Theory, Design and Synthesis |publisher = Springer |date = 2011 |pages = 372–374 |url = https://books.google.com/books?id=6W1eX4QwtyYC&pg=PA372 |isbn = 978-94-007-2189-0 |url-status = live |archive-url = https://web.archive.org/web/20171116073025/https://books.google.com/books?id=6W1eX4QwtyYC&pg=PA372&lpg=PA372&dq= |archive-date = 2017-11-16 }}</ref><ref name="Pippard">{{cite book | last = Pippard | first = A. B. | title = Response and stability: an introduction to the physical theory | publisher = CUP Archive | date = 1985 | pages = 11–12 | url = https://books.google.com/books?id=tck8AAAAIAAJ&pg=PA11 | isbn = 978-0-521-26673-4}} 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 <math display="block">\frac{\Delta v}{\Delta i} = {v \over i} = R_\text{if} < 0 </math> and thus obeys Ohm's law as if it had a negative value of resistance −''R'',<ref name="Franz" /><ref name="Hickman">{{cite book |last = Hickman |first = Ian |title = Analog Circuits Cookbook |publisher = Elsevier |date = 2013 |location = New York |pages = 8–9 |url = https://books.google.com/books?id=6__8BAAAQBAJ&q=%22negative+resistance%22&pg=PA8 |isbn = 978-1-4831-0535-2 |url-status = live |archive-url = https://web.archive.org/web/20160527193709/https://books.google.com/books?id=6__8BAAAQBAJ&pg=PA8&dq=%22negative+resistance%22 |archive-date = 2016-05-27 }}</ref> 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".<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 charge 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: *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;<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 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".<ref name="Crisson" />

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.<ref name="Ghadiri" /><ref name="Hickman" /> They are widely used in active filters<ref name="Deliyannis" /><ref name="Dimopoulos" /> because they can create transfer functions that cannot be realized with positive circuit elements.<ref name="Podell">{{cite conference | last = Podell | first = A.F. |author2=Cristal, E.G. | title = Negative-Impedance Converters (NIC) for VHF Through Microwave Circuit Applications | book-title = Microwave Symposium Digest, 1971 IEEE GMTT International 16–19 May 1971 | pages = 182–183 | publisher = Institute of Electrical and Electronics Engineers | date = May 1971 | location = USA | doi =10.1109/GMTT.1971.1122957 }} on IEEE website</ref> Examples of circuits with this type of negative resistance are the negative impedance converter (NIC), gyrator, Deboo integrator,<ref name="Dimopoulos" /><ref name="Simons">{{cite web |last = Simons |first = Elliot |title = Consider the "Deboo" integrator for unipolar noninverting designs |work = Electronic Design magazine website |publisher = Penton Media, Inc. |date = March 18, 2002 |url = http://electronicdesign.com/article/analog-and-mixed-signal/consider-the-deboo-integrator-for-unipolar-noninve |access-date = November 20, 2012 |url-status = live |archive-url = https://web.archive.org/web/20121220111355/http://electronicdesign.com/article/analog-and-mixed-signal/consider-the-deboo-integrator-for-unipolar-noninve |archive-date = December 20, 2012 }}</ref> frequency dependent negative resistance (FDNR),<ref name="Hickman" /> and generalized immittance converter (GIC).<ref name="Deliyannis" /><ref name="Karp" /><ref name="Hamilton">{{cite book |last = Hamilton |first = Scott |title = An Analog Electronics Companion: Basic Circuit Design for Engineers and Scientists |publisher = Cambridge University Press |date = 2007 |page = 528 |url = https://books.google.com/books?id=2BntAEtXsBMC&q=immittance+converter%22+%22negative+resistance&pg=PA528 |isbn = 978-0-521-68780-5 |url-status = live |archive-url = https://web.archive.org/web/20170712132224/https://books.google.com/books?id=2BntAEtXsBMC&pg=PA528&lpg=P528&dq=immittance+converter%22+%22negative+resistance |archive-date = 2017-07-12 }}</ref>

===Feedback oscillators=== If an LC circuit is connected across the input of a positive feedback amplifier like that above, the negative differential input resistance <math>R_\text{if}</math> can cancel the positive loss resistance <math>r_\text{loss}</math> inherent in the tuned circuit.<ref name="Peters">this property was often called "resistance neutralization" in the days of vacuum tubes, see {{cite journal |last = Bennett |first = Edward |author2 = Leo James Peters |title = Resistance Neutralization: An application of thermionic amplifier circuits |journal = Journal of the AIEE |volume = 41 |issue = 1 |pages = 234–248 |publisher = American Institute of Electrical Engineers |location = New York |date = January 1921 |url = https://books.google.com/books?id=TnZJAQAAIAAJ&q=%22resistance+neutralization&pg=PA234 |access-date = August 14, 2013 }} and Ch. 3: "Resistance Neutralization" in {{cite book |last = Peters |first = Leo James |title = Theory of Thermionic Vacuum Tube Circuits |publisher = McGraw-Hill |date = 1927 |pages = 62–87 |url = http://www.tubebooks.org/Books/peters_theory.pdf |url-status = live |archive-url = https://web.archive.org/web/20160304043123/http://www.tubebooks.org/Books/peters_theory.pdf |archive-date = 2016-03-04 }}</ref> If <math>R_\text{if}\;=\;-r_\text{loss}</math> this will create in effect a tuned circuit with zero AC resistance (poles on the ''jω'' axis).<ref name="Solymar" /><ref name="Armstrong" /> 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.<ref name="Prasad" /><ref name="Lee">{{cite book | last = Lee | first = Thomas H. | title = The Design of CMOS Radio-Frequency Integrated Circuits, 2nd Ed. | publisher = Cambridge University Press | date = 2004 | location = UK | pages = 641–642 | url = https://books.google.com/books?id=io1hL48OqBsC&q=%22negative+impedance+converter&pg=PA641 | isbn = 978-0-521-83539-8}}</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 |last = Kung |first = Fabian Wai Lee |title = Lesson 9: Oscillator Design |work = RF/Microwave Circuit Design |publisher = Prof. Kung's website, Multimedia University |date = 2009 |url = http://pesona.mmu.edu.my/~wlkung/ADS/rf/lesson9.pdf |access-date = October 17, 2012 |archive-url = https://web.archive.org/web/20150722165131/http://pesona.mmu.edu.my/~wlkung/ADS/rf/lesson9.pdf |archive-date = July 22, 2015 }}, 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">[https://books.google.com/books?id=e_oZ69GAuxAC&dq=%22negative-resistance%22&pg=PA84 Gottlieb 1997, ''Practical Oscillator Handbook'', p. 84] {{webarchive|url=https://web.archive.org/web/20160515053022/https://books.google.com/books?id=e_oZ69GAuxAC |date=2016-05-15 }}</ref>

===Q enhancement=== A tuned circuit connected to a negative resistance which cancels some but not all of its parasitic loss resistance (so <math>|R_\text{if}|\;<\;r_\text{loss}</math>) will not oscillate, but the negative resistance will decrease the damping in the circuit (moving its poles toward the ''jω'' axis), increasing its Q factor so it has a narrower bandwidth and more selectivity.<ref name="Peters" /><ref name="Li">{{cite conference | first = Dandan | last = Li |author2=Yannis Tsividis | title = Active filters using integrated inductors | book-title = Design of High Frequency Integrated Analogue Filters | page = 58 | publisher = Institution of Engineering and Technology (IET) | date = 2002 | url = https://books.google.com/books?id=SYKtbXYqrAoC&pg=PA58 | isbn = 0-85296-976-7 | access-date = July 23, 2013}}</ref><ref name="Rembovsky">{{cite book |last = Rembovsky |first = Anatoly |title = Radio Monitoring: Problems, Methods and Equipment |publisher = Springer |date = 2009 |page = 24 |url = https://books.google.com/books?id=2ra1lg9MCLgC&q=%22negative+resistance&pg=PA24 |isbn = 978-0-387-98100-0 |url-status = live |archive-url = https://web.archive.org/web/20170719144717/https://books.google.com/books?id=2ra1lg9MCLgC&pg=PA24&dq=%22negative+resistance |archive-date = 2017-07-19 }}</ref><ref name="Sun">{{cite book | last = Sun | first = Yichuang Sun | title = Design of High Frequency Integrated Analogue Filters | publisher = IET | date = 2002 | pages = 58, 60–62 | url = https://books.google.com/books?id=SYKtbXYqrAoC&pg=PA58 | isbn = 978-0-85296-976-2}}</ref> Q enhancement, also called ''regeneration'', was first used in the regenerative radio receiver invented by Edwin Armstrong in 1912<ref name="Armstrong" /><ref name="Rembovsky" /> and later in "Q multipliers".<ref name="Carr2">{{cite book | last = Carr | first = Joseph | title = Antenna Toolkit, 2nd Ed. | publisher = Newnes | date = 2001 | page = 193 | url = https://books.google.com/books?id=kEbQ3io1q6sC&pg=PA193 | isbn = 978-0-08-049388-6}}</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" />

===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.<ref name="Kennedy">{{cite journal |last = Kennedy |first = Michael Peter |title = Three Steps to Chaos: Part 1 – Evolution |journal = IEEE Transactions on Circuits and Systems |volume = 40 |issue = 10 |page = 640 |date = October 1993 |url = http://www.eecs.berkeley.edu/~chua/papers/Kennedy93.pdf |doi = 10.1109/81.246140 |access-date = February 26, 2014 |url-status = live |archive-url = https://web.archive.org/web/20131105084249/http://www.eecs.berkeley.edu/~chua/papers/Kennedy93.pdf |archive-date = November 5, 2013 }}</ref> 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.<ref name="Kennedy" /> This is usually synthesized using a negative impedance converter circuit.<ref name="Kennedy" />

===Negative impedance converter=== {{multiple image | align = right | direction = horizontal | header = | image1 = General negative impedance circuit.svg | width1 = 150 | image2 = Negative impedance converter IV curve.svg | width2 = 200 | caption2 = 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)<ref name="HorowitzVideo" /><ref name="Hickman" /><ref name="Lee" /><ref name="Linvill">{{cite journal |last = Linvill |first = J.G. |title = Transistor Negative-Impedance Converters |journal= Proceedings of the IRE |pages = 725–729 |year = 1953 |doi = 10.1109/JRPROC.1953.274251 |volume = 41 |issue = 6|bibcode = 1953PIRE...41..725L |s2cid = 51654698 }}</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 <math display="block">v_o = v(R_1 + R_1)/R_1 = 2v </math> 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 <math display="block">i = \frac {v - v_o}{Z} = \frac {v - 2v}{Z} = - \frac {v}{Z} </math> So the input impedance to the circuit is<ref name="Muthuswamy" /> <math display="block">z_\text{in} = \frac {v}{i} = -Z </math> 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 | title = Application Note 1868: Negative resistor cancels op-amp load | work = Application Notes | publisher = Maxim Integrated, Inc. website | date = January 31, 2003 | url = http://www.maximintegrated.com/en/app-notes/index.mvp/id/1868 | access-date = October 8, 2014}}</ref> for example they were originally developed to cancel resistance in telephone cables, serving as repeaters.<ref name="Lee" />

===Negative capacitance and inductance=== 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 impedance <math>Z_\text{C}(j\omega)</math> of <math display="block">i = -C {dv \over dt} \qquad\qquad Z_C = -1/j\omega C</math> 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 <math display="block">v = -L {di \over dt} \qquad\qquad Z_L = -j\omega L </math> 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 Foster's reactance theorem.<ref name="Hansen">{{cite book | last = Hansen | first = Robert C. |author2=Robert E. Collin | title = Small Antenna Handbook | publisher = John Wiley & Sons | date = 2011 | pages = sec. 2–6, pp. 262–263 | url = https://books.google.com/books?id=Qmkqeke3dzAC&pg=PA262 | isbn = 978-0-470-89083-7}}</ref> 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.<ref name="Aberle">{{cite book |last = Aberle |first = James T. |author2 = Robert Loepsinger-Romak |title = Antennas With Non-Foster Matching Networks |publisher = Morgan & Claypool |date = 2007 |pages = 1–8 |url = https://books.google.com/books?id=4jt4gBgiDbIC&pg=PA5 |isbn = 978-1-59829-102-5 |url-status = live |archive-url = https://web.archive.org/web/20171017154929/https://books.google.com/books?id=4jt4gBgiDbIC&pg=PA5 |archive-date = 2017-10-17 }}</ref> This would allow the creation of small compact antennas that would have broad bandwidth,<ref name="Aberle" /> exceeding the Chu–Harrington limit.

==Oscillators== [[File:Ganna gjenerators M31102-1.jpg|thumb|upright=0.8|An oscillator consisting of a Gunn diode inside a cavity resonator. The negative resistance of the diode excites microwave oscillations in the cavity, which radiate through the aperture into a waveguide ''(not shown)''. ]]

Negative differential resistance devices are widely used to make electronic oscillators.<ref name="Carr" /><ref name="Rybin" /><ref name="Haddad">{{cite conference | first = G. I. | last = Haddad |author2=J. R. East |author3=H. Eisele | title = Two-terminal active devices for terahertz sources | book-title = Terahertz Sensing Technology: Electronic devices and advanced systems technology | page = 45 | publisher = World Scientific | date = 2003 | url = https://books.google.com/books?id=JVrgksZq-zgC&q=%22negative+differential+resistance%22+oscillator+resonator&pg=PA45 | isbn = 978-981-279-682-0 | access-date = October 17, 2012}}</ref> 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<ref name="Räisänen">{{cite book |last = Räisänen |first = Antti V. |author2 = Arto Lehto |title = Radio Engineering for Wireless Communication and Sensor Applications |publisher = Artech House |date = 2003 |location = USA |pages = 180–182 |url = https://books.google.com/books?id=m8Dgkvf84xoC&pg=PA181 |isbn = 978-1-58053-542-7 |url-status = live |archive-url = https://web.archive.org/web/20170225055401/https://books.google.com/books?id=m8Dgkvf84xoC&pg=PA181 |archive-date = 2017-02-25 }}</ref> with a DC source to bias the device into its negative resistance region and provide power.<ref name="Laplante">{{cite book | last = Laplante | first = Philip A. Laplante | title = Comprehensive Dictionary of Electrical Engineering, 2nd Ed. | publisher = CRC Press | date = 2005 | page = 466 | url = https://books.google.com/books?id=_UBzZ4coYMkC&q=%22negative+resistance+oscillator%22+%22negative+differential+resistance&pg=PA466 | isbn = 978-0-8493-3086-5}}</ref><ref name="Chen3">{{cite book |last = Chen |first = Wai Kai |title = The Electrical Engineering Handbook |publisher = Academic Press |date = 2004 |location = London |page = 698 |url = https://books.google.com/books?id=qhHsSlazGrQC&q=%22negative+resistance+oscillator&pg=PA698 |isbn = 978-0-12-170960-0 |url-status = live |archive-url = https://web.archive.org/web/20160819081609/https://books.google.com/books?id=qhHsSlazGrQC |archive-date = 2016-08-19 }}</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 resonant frequency.<ref name="Lesurf" /><ref name="Solymar" />

===Uses=== Negative resistance oscillators are mainly used at high frequencies in the microwave range or above, since feedback oscillators 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 radar speed guns, and local oscillators for satellite receivers. They are a widely used source of microwave energy, and virtually the only solid-state source of millimeter wave<ref name="Du2">{{cite book | last = Du | first = Ke-Lin |author2=M. N. S. Swamy | title = Wireless Communication Systems: From RF Subsystems to 4G Enabling Technologies | publisher = Cambridge University Press | date = 2010 | page = 438 | url = https://books.google.com/books?id=5dGjKLawsTkC&pg=PA438 | isbn = 978-0-521-11403-5}}</ref> and terahertz energy<ref name="Haddad" /> Negative resistance microwave vacuum tubes such as magnetrons produce higher power outputs,<ref name="Räisänen" /> in such applications as radar transmitters and microwave ovens. Lower frequency relaxation oscillators can be made with UJTs and gas-discharge lamps such as neon lamps.

The negative resistance oscillator model is not limited to one-port devices like diodes but can also be applied to feedback oscillator circuits with two port devices such as transistors and tubes.<ref name="Kung" /><ref name="Räisänen" /><ref name="Ellinger">{{cite book |last = Ellinger |first = Frank |title = Radio Frequency Integrated Circuits and Technologies, 2nd Ed. |publisher = Springer |date = 2008 |location = USA |pages = 391–394 |url = https://books.google.com/books?id=0pl9xYD0QNMC&pg=PA391 |isbn = 978-3-540-69324-6 |url-status = live |archive-url = https://web.archive.org/web/20160731222206/https://books.google.com/books?id=0pl9xYD0QNMC&pg=PA391&dq= |archive-date = 2016-07-31 }}</ref><ref name="Gottlieb3">{{cite book |last = Gottlieb |first = Irving M. |title = Practical Oscillator Handbook |publisher = Elsevier |date = 1997 |pages = 84–85 |url = https://books.google.com/books?id=e_oZ69GAuxAC&q=%22negative+resistance%22+%22feedback&pg=PA84 |isbn = 978-0-08-053938-6 |url-status = live |archive-url = https://web.archive.org/web/20160515053022/https://books.google.com/books?id=e_oZ69GAuxAC |archive-date = 2016-05-15 }}</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=== {{multiple image | align = right | direction = horizontal | image1 = Gunn diode oscillator circuit.svg | caption1 = Gunn diode oscillator circuit | width1 = 150 | image2 = Gunn diode oscillator AC circuit.svg | caption2 = AC equivalent circuit | width2 = 102 | footer = }} {{main | Gunn oscillator }}

[[File:Negative resistance oscillator load lines.svg|thumb|upright=1.3|Gunn diode oscillator load lines. ''DCL'': DC load line, which sets the Q point. ''SSL'': negative resistance during startup while amplitude is small. Since <math>r\;<\;R</math> 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 <math>r</math> decreases until it equals <math>R</math>. ]]

The common Gunn diode 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 choke ''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 Kirchhoff's Voltage Law gives a differential equation for <math>i(t)</math>, the AC current<ref name="Lesurf" /> <math display="block">\frac {d^2 i}{dt^2} + \frac {R - r}{L}\frac {di}{dt} + \frac {1}{LC}i = 0 </math> Solving this equation gives a solution of the form<ref name="Lesurf" /> <math display="block">i(t) = i_0 e^{\alpha t} \cos(\omega t + \phi) </math> where <math display="block">\alpha = \frac{r - R}{2L} \quad \omega = \sqrt{\frac {1}{LC} - \left(\frac {r - R}{2L}\right)^2} </math> This shows that the current through the circuit, <math>i(t)</math>, varies with time about the DC Q point, <math>I_\text{bias}</math>. When started from a nonzero initial current <math>i(t) = i_0</math> the current oscillates sinusoidally at the resonant frequency '''''ω''''' of the tuned circuit, with amplitude either constant, increasing, or decreasing exponentially, depending on the value of '''''α'''''. Whether the circuit can sustain steady oscillations depends on the balance between <math>R</math> and <math>r</math>, the positive and negative resistance in the circuit:<ref name="Lesurf" /> #left|80px<math>r < R \Rightarrow \alpha < 0</math>: (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 <math>R</math> and die away exponentially to zero, as in an ordinary tuned circuit.<ref name="Solymar" /> So the circuit does not oscillate. #left|80px<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. #left|80px<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 |date = 2009 |url = http://pesona.mmu.edu.my/~wlkung/ADS/rf/lesson9.pdf |access-date = October 17, 2012 |archive-url = https://web.archive.org/web/20120526153220/http://pesona.mmu.edu.my/~wlkung/ADS/rf/lesson9.pdf |archive-date = May 26, 2012 }}, Sec. 3 Negative Resistance Oscillators, p. 21</ref> When the power is turned on, electrical noise in the circuit provides a signal <math>i_0</math> to start spontaneous oscillations, which grow exponentially. However, the oscillations cannot grow forever; the nonlinearity of the diode eventually limits the amplitude.

At large amplitudes the circuit is nonlinear, so the linear analysis above does not strictly apply and differential resistance is undefined; but the circuit can be understood by considering <math>r</math> to be the "average" resistance over the cycle. As the amplitude of the sine wave exceeds the width of the negative resistance region and the voltage swing extends into regions of the curve with positive differential resistance, the average negative differential resistance <math>r</math> becomes smaller, and thus the total resistance <math>R\;-\;r</math> and the damping <math>\alpha</math> becomes less negative and eventually turns positive. Therefore, the oscillations will stabilize at the amplitude at which the damping becomes zero, which is when <math>r\;=\;R</math>.<ref name="Lesurf" />

Gunn diodes have negative resistance in the range −5 to −25 ohms.<ref name="Kshetrimayum">{{cite web |last = Kshetrimayum |first = Rakhesh Singh |title = Experiment 5: Study of ''I–V'' Characteristics of Gunn Diodes |work = EC 341 Microwave Laboratory |publisher = Electrical Engineering Dept., Indian Institute of Technology, Guwahati, India |url = http://www.iitg.ernet.in/engfac/krs/public_html/lab/ee442/Exp5.pdf |access-date = January 8, 2013 |url-status = live |archive-url = https://web.archive.org/web/20140124181833/http://www.iitg.ernet.in/engfac/krs/public_html/lab/ee442/Exp5.pdf |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=== 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 |last = Rhea |first = Randall W. |title = Discrete Oscillator Design: Linear, Nonlinear, Transient, and Noise Domains |publisher = Artech House |date = 2010 |location = USA |pages = 57, 59 |url = https://books.google.com/books?id=4Op56QdHFPUC&pg=PA57 |isbn = 978-1-60807-047-3 |url-status = live |archive-url = https://web.archive.org/web/20171011030329/https://books.google.com/books?id=4Op56QdHFPUC&pg=PA57&lpg=PA57 |archive-date = 2017-10-11 }}</ref><ref name="Krugman">{{cite book |last = Krugman |first = Leonard M. |title = Fundamentals of Transistors |publisher = John F. Rider |date = 1954 |location = New York |pages = 101–102 |url = http://www.vias.org/transistor_basics/transistor_basics_06_03_03.html |url-status = live |archive-url = https://web.archive.org/web/20140819125558/http://www.vias.org/transistor_basics/transistor_basics_06_03_03.html |archive-date = 2014-08-19 }} reprinted on [http://www.vias.org/about.html Virtual Institute of Applied Science] {{webarchive|url=https://web.archive.org/web/20141223065727/http://www.vias.org/about.html |date=2014-12-23 }} website</ref> right|160px *'''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''bias to bias the device into its negative resistance region, and parallel resonant circuit load ''LC''. The resonant circuit has high impedance only at its resonant frequency, so the circuit will be unstable and oscillate only at that frequency. {{clear}} right|160px *'''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''bias (which may consist of a voltage source in series with a large resistor) and series resonant circuit ''LC''. The series LC circuit has low impedance only at its resonant frequency and so will only oscillate there. {{clear}}

===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.<ref name="Maas" /><ref name="Ellinger" /><ref name="Kurokawa">{{cite journal | last = Kurokawa | first = Kaneyuki | title = Some Basic Characteristics of Broadband Negative Resistance Oscillator Circuits | journal = Bell System Tech. J. | volume = 48 | issue = 6 | pages = 1937–1955 | date = July 1969 | url = https://archive.org/details/bstj48-6-1937 | doi = 10.1002/j.1538-7305.1969.tb01158.x | bibcode = 1969BSTJ...48.1937K | 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''" ''(red)'' 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 |last = Rohde |first = Ulrich L. |author2 = Ajay K. Poddar |author3 = Georg Böck |title = The Design of Modern Microwave Oscillators for Wireless Applications:Theory and Optimization |publisher = John Wiley & Sons |date = 2005 |location = USA |pages = 96–97 |url = https://books.google.com/books?id=GrvgJe8aujcC&pg=PA96 |isbn = 978-0-471-72716-3 |url-status = live |archive-url = https://web.archive.org/web/20170921060746/https://books.google.com/books?id=GrvgJe8aujcC&pg=PA96 |archive-date = 2017-09-21 }}</ref> The complex impedance of the negative resistance part <math>Z_N = R_N(I, \omega) + jX_N(I, \omega) </math> depends on frequency ''ω'' but is also nonlinear, in general declining with the amplitude of the AC oscillation current ''I''; while the resonator part <math>Z_L = R_L(\omega) + jX_L(\omega) </math> is linear, depending only on frequency.<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 conjugate<ref name="Frank" /><ref name="Räisänen" /><ref name="Ellinger" /><ref name="Rohde" /> 300px|right

<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" />

Alternately, the condition for oscillation can be expressed using the reflection coefficient.<ref name="Frank" /> The voltage waveform at the reference plane can be divided into a component ''V''1 travelling toward the negative resistance device and a component ''V''2 travelling in the opposite direction, toward the resonator part. The reflection coefficient of the active device <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" /> <math display="block">|\Gamma_N \Gamma_L| \ge 1 </math> 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,<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== 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.<ref name="Golio2" /> 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.<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 nonreciprocal devices such as isolators and directional couplers.<ref name="Golio2" />

===Reflection amplifier=== {{multiple image | align = right | direction = vertical | header = | image1 = Reflection amplifier.svg | caption1 = AC equivalent circuit of reflection amplifier | width1 = 250 | image2 = 10Gig Tunnel Amp M.jpg | caption2 = 8–12 GHz microwave amplifier consisting of two cascaded tunnel diode reflection amplifiers | width2 = 250 | footer = }}

One widely used circuit is the ''reflection amplifier'' in which the separation is accomplished by a ''circulator''.<ref name="Golio2" /><ref name="Das">{{cite book | last1 = Das | first1 = Annapurna | last2 = Das | first2 = Sisir K. | title = Microwave Engineering | publisher = Tata McGraw-Hill Education | date = 2000 | pages = 394–395 | url = https://books.google.com/books?id=ZU19Uemy83YC&q=%22reflection+amplifier%22negative+resistance&pg=PA394 | isbn = 978-0-07-463577-3 }}</ref><ref name="Willardson">H. C. Okean, ''Tunnel diodes'' in {{cite book | last1 = Willardson | first1 = Robert K. | last2 = Beer | first2 = Albert C., Eds. | title = Semiconductors and Semimetals, Vol. 7 Part B | publisher = Academic Press | date = 1971 | pages = 546–548 | url = https://books.google.com/books?id=TbtPwOVDcegC&q=%22reflection+amplifier%22negative+resistance&pg=PA547 | isbn = 978-0-08-086397-9 }}</ref><ref name="Button">Chang, Kai, ''Millimeter-wave Planar Circuits and Subsystems'' in {{cite book | last1 = Button | first1 = Kenneth J., Ed. | title = Infrared and Millimeter Waves: Millimeter Components and Techniques, Part 5 | publisher = Academic Press | volume = 14 | date = 1985 | pages = 133–135 | url = https://books.google.com/books?id=rgSGCo3qiZgC&q=%22reflection+amplifier&pg=PA134 | isbn = 978-0-323-15061-3 }}</ref> A circulator is a nonreciprocal solid-state component with three ports (connectors) which transfers a signal applied to one port to the next in only one direction, port 1 to port 2, 2 to 3, and 3 to 1. In the reflection amplifier diagram the input signal is applied to port 1, a biased VCNR negative resistance diode ''N'' is attached through a filter ''F'' to port 2, and the output circuit is attached to port 3. The input signal is passed from port 1 to the diode at port 2, but the outgoing "reflected" amplified signal from the diode is routed to port 3, so there is little coupling from output to input. The characteristic impedance <math>Z_0</math> of the input and output transmission lines, usually 50Ω, is matched to the port impedance of the circulator. The purpose of the filter ''F'' is to present the correct impedance to the diode to set the gain. At radio frequencies NR diodes are not pure resistive loads and have reactance, so a second purpose of the filter is to cancel the diode reactance with a conjugate reactance to prevent standing waves.<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 <math display="block">P_\text{in} = V_I^2 / R_1</math> The output power from the diode is <math display="block">P_\text{out} = V_R^2 / R_1</math> So the power gain <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 |last1 = Linkhart |first1 = Douglas K. |title = Microwave Circulator Design |publisher = Artech House |edition = 2 |date = 2014 |pages = 78–81 |url = https://books.google.com/books?id=AutPAwAAQBAJ&q=circulator&pg=PA79 |isbn = 978-1-60807-583-6 |url-status = live |archive-url = https://web.archive.org/web/20171210183529/https://books.google.com/books?id=AutPAwAAQBAJ&pg=PA79&dq=circulator |archive-date = 2017-12-10 }}</ref> <math display="block">G_\text{P} = {P_\text{out} \over P_\text{in}} = {V_R^2 \over V_I^2} = |\Gamma|^2</math>

<math display="block">|\Gamma|^2 = \left|{Z_N - Z_1 \over Z_N + Z_1}\right|^2</math> <math display="block">|\Gamma|^2 = \left|{R_N + jX_N - (R_1 + jX_1)\over R_N + jX_N + R_1 + jX_1}\right|^2</math> <math>R_\text{N}</math> is the negative resistance of the diode '''−''r'''''. Assuming the filter is matched to the diode so <math>X_1 = -X_N</math><ref name="Button" /> then the gain is <math display="block">G_\text{P} = |\Gamma|^2 = {(r + R_1)^2 + 4X_N^2 \over (r - R_1)^2} </math> 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.

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.<ref name="Linkhart" />

==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.<ref name="Kumar2" /> 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,<ref name="Abraham" /> 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.<ref name="MacLean">{{cite journal | last1 = MacLean | first1 = Jason N. | last2 = Schmidt | first2 = Brian J. | title = Voltage-Sensitivity of Motoneuron NMDA Receptor Channels Is Modulated by Serotonin in the Neonatal Rat Spinal Cord | journal = Journal of Neurophysiology | volume = 86 | issue = 3 | pages = 1131–1138 | date = September 2001 | doi =10.1152/jn.2001.86.3.1131 | pmid = 11535663 | s2cid = 8074067 }}</ref> 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.<ref name="Hong">{{cite book |last = Hong |first = Sungook |title = Wireless: From Marconi's Black-Box to the Audion |publisher = MIT Press |date = 2001 |location = USA |pages = 159–165 |url = http://monoskop.org/images/f/f4/Hong_Sungook_Wireless_From_Marconis_Black-Box_to_the_Audion.pdf |isbn = 978-0-262-08298-3 |url-status = live |archive-url = https://web.archive.org/web/20140819090610/http://monoskop.org/images/f/f4/Hong_Sungook_Wireless_From_Marconis_Black-Box_to_the_Audion.pdf |archive-date = 2014-08-19 }}</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 | last = Heaviside | first = Oliver | title = Correspondence: Negative Resistance | journal = The Electrician | volume = 37 | issue = 14 | page = 452 | publisher = "The Electrician" Printing and Publishing Co. | location = London | date = July 31, 1892 | url = https://books.google.com/books?id=PAJRAAAAYAAJ&pg=PA452 | access-date = December 24, 2012}}, also see letter by Andrew Gray on same page</ref> 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.<ref name="Ayrton">{{cite journal | last = Ayrton | first = Hertha | title = The Mechanism of the Electric Arc | journal = The Electrician | volume = 47 | issue = 17 | pages = 635–636 | publisher = The Electrician Printing & Publishing Co. | location = London | date = August 16, 1901 | url = https://books.google.com/books?id=TQ1RAAAAYAAJ&q=%22negative+resistance&pg=PA635 | access-date = January 2, 2013}}</ref><ref name="Britannica" /><ref name="Gethemann">{{cite web |last = Gethemann |first = Daniel |title = Singing Arc: The Usefulness of Negative Resistance |work = Zauberhafte Klangmaschinen |publisher = Institut fur Medienarchaologie |date = 2012 |url = http://klangmaschinen.ima.or.at/db/db.php?id=37&table=Object&lang=en&showartikel=1&view=ausstellung |access-date = 2012-04-11 |url-status = live |archive-url = https://web.archive.org/web/20120104062445/http://klangmaschinen.ima.or.at/db/db.php?id=37&table=Object&lang=en&showartikel=1&view=ausstellung |archive-date = 2012-01-04 }}</ref> Frith and Rodgers in 1896<ref name="Britannica" /><ref name="Frith">{{cite journal | last = Frith | first = Julius |author2=Charles Rodgers | title = On the Resistance of the Electric Arc | journal = London, Edinburgh, and Dublin Philosophical Magazine | volume = 42 | issue = 258 | pages = 407–423 | date = November 1896 | url = https://books.google.com/books?id=snw7AQAAMAAJ&pg=PA407 | doi =10.1080/14786449608620933 | 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 Institute of Electrical Engineers.<ref name="Gethemann" />

===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.<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 = Joseph, Ed. |title = The Scientific Writings of the late George Francis Fitzgerald |publisher = Longmans, Green and Co. |date = 1902 |location = London |pages = 277–281 |url = https://books.google.com/books?id=G0bPAAAAMAAJ&pg=PA277 |url-status = live |archive-url = https://web.archive.org/web/20140707134922/https://books.google.com/books?id=G0bPAAAAMAAJ&pg=PA277 |archive-date = 2014-07-07 }}</ref> In the same year Elihu Thomson built a negative resistance oscillator by connecting an LC circuit to the electrodes of an arc,<ref name="Nahin">{{cite book |last = Nahin |first = Paul J. |title = The Science of Radio: With Matlab and Electronics Workbench Demonstration, 2nd Ed. |publisher = Springer |date = 2001 |pages = 81–85 |url = https://books.google.com/books?id=V1GBW6UD4CcC&q=%22van+der+pol%22+%22negative+resistance%22+nonlinear&pg=PA82 |isbn = 978-0-387-95150-8 |url-status = live |archive-url = https://web.archive.org/web/20170225070713/https://books.google.com/books?id=V1GBW6UD4CcC&pg=PA82&lpg=PA82&dq=%22van+der+pol%22+%22negative+resistance%22+nonlinear |archive-date = 2017-02-25 }}</ref><ref name="Morse">{{cite book |last = Morse |first = A. H. |title = Radio: Beam and Broadcast |publisher = Ernest Benn |date = 1925 |location = London |page = 28 |url = https://archive.org/stream/radiobeamandbroa029214mbp#page/n27/mode/2up |url-status = live |archive-url = https://web.archive.org/web/20160315213300/https://archive.org/stream/radiobeamandbroa029214mbp#page/n27/mode/2up |archive-date = 2016-03-15 }}</ref> 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.<ref name="Nahin" /><ref name="Hong" /><ref name="Gethemann" /> 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.<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 "singing arc" oscillator was limited to audio frequencies.<ref name="Nahin" /> 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,<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 Poulsen arc radio transmitter, which was widely used until the 1920s.<ref name="Nahin" /><ref name="Hong" />

===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" /> Heinrich Barkhausen in 1907 showed that oscillators must have negative resistance.<ref name="Duncan" /> 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.<ref name="Hong" />

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.<ref name="Gottlieb" /><ref name="Butler" /><ref name="Hull">{{cite journal | last = Hull | first = Albert W. | title = The Dynatron – A vacuum tube possessing negative electric resistance | journal = Proceedings of the IRE | volume = 6 | issue = 1 | pages = 5–35 | date = February 1918 | url = https://books.google.com/books?id=IUASAAAAIAAJ&q=hull+dynatron&pg=PA5 | doi = 10.1109/jrproc.1918.217353 | bibcode = 1918PIRE....6....5H | s2cid = 51656451 | access-date = 2012-05-06| url-access = subscription }}</ref> 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.<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 dynatron oscillator. Other negative resistance tube oscillators followed, such as the magnetron 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 | last = Latour | first = Marius | title = Basic Theory of Electron-Tube Amplifiers – Part II | journal = Electrical World | volume = 76 | issue = 18 | pages = 870–872 | publisher = McGraw-Hill | location = New York | date = October 30, 1920 | url = https://books.google.com/books?id=aedQAAAAYAAJ&q=%22negative+capacitance%22&pg=PA872 | access-date = December 27, 2012}}</ref><ref name="Merrill">{{cite journal | last = Merrill | first = J.L. Jr. | title = Theory of the Negative Impedance Converter | journal = Bell System Tech. J. | volume = 30 | issue = 1 | pages = 88–109 | date = January 1951 | url = https://archive.org/details/bstj30-1-88 | doi = 10.1002/j.1538-7305.1951.tb01368.x | bibcode = 1951BSTJ...30...88M | 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 repeaters at Bell Labs 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=== 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<ref name="Grebennikov">{{cite book |last = Grebennikov |first = Andrei |title = RF and Microwave Transmitter Design |publisher = John Wiley & Sons |date = 2011 |page = 4 |url = https://books.google.com/books?id=nGLdHfULzhYC&q=%22negative+resistance%22++%22crystal+detector%22&pg=PA4 |isbn = 978-0-470-52099-4 |url-status = live |archive-url = https://web.archive.org/web/20160917100859/https://books.google.com/books?id=nGLdHfULzhYC&pg=PA4&dq=%22negative+resistance%22++%22crystal+detector%22&hl=en#v=onepage&q=%22negative%20resistance%22%20%20%22crystal%20detector%22&f=false |archive-date = 2016-09-17 }}</ref><ref name="Pickard">{{cite journal | last = Pickard | first = Greenleaf W. | title = The Discovery of the Oscillating Crystal | journal = Radio News | volume = 6 | issue = 7 | page = 1166 | publisher = Experimenter Publishing Co. | location = New York |date=January 1925 | url = https://www.worldradiohistory.com/Archive-Radio-News/20s/Radio-News-1925-01-R.pdf | access-date = July 15, 2014}}</ref> and G. W. Pickard.<ref name="Pickard" /><ref name="White">{{cite web |last = White |first = Thomas H. |title = Section 14 – Expanded Audio and Vacuum Tube Development (1917–1930) |work = United States Early Radio History |publisher = earlyradiohistory.us |date = 2021 |url = https://earlyradiohistory.us/sec014.htm |access-date = May 5, 2021 }}</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 Oleg Losev, who in 1922 discovered negative differential resistance in biased zincite (zinc oxide) point contact junctions.<ref name="White" /><ref name="Losev">{{cite journal | last = Losev | first = O. V. | title = Oscillating Crystals | journal = Radio News | volume = 6 | issue = 7 | pages = 1167, 1287 | publisher = Experimenter Publishing Co. | location = New York |date=January 1925 | url = https://www.worldradiohistory.com/Archive-Radio-News/20s/Radio-News-1925-01-R.pdf | access-date = July 15, 2014}}</ref><ref name="Gabel">{{cite journal |last = Gabel |first = Victor |title = The Crystal as a Generator and Amplifier |journal = The Wireless World and Radio Review |volume = 15 |pages = 2–5 |publisher = Iliffe & Sons Ltd. |location = London |date = October 1, 1924 |url = http://www.hpfriedrichs.com/downloads-lib/xtalgen.pdf |access-date = March 20, 2014 |url-status = live |archive-url = https://web.archive.org/web/20141023072450/http://www.hpfriedrichs.com/downloads-lib/xtalgen.pdf |archive-date = October 23, 2014 }}</ref><ref name="Ben-Menahem">{{cite book |last = Ben-Menahem |first = Ari |title = Historical Encyclopedia of Natural and Mathematical Sciences, Vol. 1 |publisher = Springer |date = 2009 |page = 3588 |url = https://books.google.com/books?id=9tUrarQYhKMC&q=losev+%22negative+resistance%22&pg=PA3588 |isbn = 978-3-540-68831-0 |url-status = live |archive-url = https://web.archive.org/web/20171123190123/https://books.google.com/books?id=9tUrarQYhKMC&pg=PA3588&dq=losev+%22negative+resistance%22&hl=en&sa=X&ei=EKa8T4LxL8fiiAKm4IHEDQ&ved=0CEAQ6AEwAg#v=onepage&q=losev%20%22negative%20resistance%22&f=false |archive-date = 2017-11-23 }}</ref><ref name="Lee2">[https://books.google.com/books?id=io1hL48OqBsC&pg=PA20 Lee, Thomas H. (2004) The Design of CMOS Radio-Frequency Integrated Circuits, 2nd Ed., p. 20]</ref> He used these to build solid-state amplifiers, oscillators, and amplifying and regenerative radio receivers, 25 years before the invention of the transistor.<ref name="Grebennikov" /><ref name="Gabel" /><ref name="Lee2" /><ref name="Gernsback">{{cite journal |last = Gernsback |first = Hugo |title = A Sensational Radio Invention |journal = Radio News |page = 291 |publisher = Experimenter Publishing |date = September 1924 |url = https://books.google.com/books?id=2rQ1AQAAIAAJ&pg=PA291 |access-date = May 5, 2021 }} and "[https://books.google.com/books?id=2rQ1AQAAIAAJ&pg=PA294 The Crystodyne Principle]", pp. 294–295</ref> Later he even built a superheterodyne receiver.<ref name="Lee2" /> 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),<ref name="Gernsback" /> and it was forgotten.<ref name="Lee2" />

The first widely used solid-state negative resistance device was the tunnel diode, invented in 1957 by Japanese physicist Leo Esaki.<ref name="Franz" /><ref name="Esaki">{{cite journal | last1 = Esaki | first1 = Leo | title = New Phenomenon in Narrow Germanium p−n Junctions | journal = Physical Review | volume = 109 | issue =2 | pages = 603–604 | date = January 1958 | doi = 10.1103/PhysRev.109.603 |bibcode = 1958PhRv..109..603E }}</ref> 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,<ref name="Ridley">{{cite journal | last = Ridley | first = B. K. | title = "Electric bubbles" and the quest for negative resistance | journal = New Scientist | volume = 22 | issue = 390 | pages = 352–355 | publisher = Cromwell House | location = London | date = May 7, 1964 | url = https://books.google.com/books?id=Bk7nTSxPE3gC&q=negative+resistance&pg=PA354 | access-date = November 15, 2012 }}{{Dead link|date=August 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> 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.<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== {{Reflist|group=note}}

==References== {{Reflist|2}}

==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 = 978-0-08-053938-6}} 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 = 978-0-262-08298-3}}, 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 = https://www.accessscience.com/content/negative-resistance-circuits/446710 | access-date = May 17, 2012| url-access = subscription }} Elementary one-page introduction to negative resistance.

Category:Electricity Category:Electronics concepts Category:Microwave technology Category:Physical quantities