Editing Negative resistance (section)

==Definitions==
[[File:DifferentialChordalResistance.svg|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 [[electrical resistance|resistance]] between two terminals of an electrical device or circuit is determined by its current–voltage (''I–V'') curve ([[current–voltage characteristic|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 = 978-0766820838}}</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 [[linear circuit|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        = 978-0205083770
 |url-status  = dead
 |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        = 978-9814317009
 |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
 |url-status  = dead
 |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 ''(<span style="color:red;">red</span>)'']]

*'''Static resistance''' (also called ''chordal resistance'', ''absolute resistance'' or just ''resistance'') – This is the common definition of resistance; the voltage divided by the current:<ref name="Aluf" /><ref name="Herrick" /><ref name="Simin" /> <math display="block">R_\mathrm{static} = \frac{v}{i} .</math> It is the inverse slope of the line ([[chord (geometry)|chord]]) from the origin through the point on the ''I–V'' curve.<ref name="Shanefield" /> In a power source, like a [[battery (electricity)|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 = 978-0853126065
 }}</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 = 978-1118498033}}</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        = 978-8184316971
 |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        = 978-9048194421
 |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 [[electric power|power dissipation]] in a component.<ref name="Traylor" /><ref name="Bakshi" /> [[Passivity (engineering)|Passive]] devices, which consume electric power, have positive static resistance; while [[passivity (engineering)|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 = 978-1118038239}} 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 (mathematics)|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 [[ohm]]s.

[[Electrical conductance|Conductance]] is the [[reciprocal (mathematics)|reciprocal]] of [[electrical resistance|resistance]].<ref name="Herrick2">{{cite book
 |last1       = Herrick
 |first1      = Robert J.
 |title       = DC/AC Circuits and Electronics: Principles & Applications
 |publisher   = Cengage Learning
 |date        = 2003
 |pages       = 105
 |url         = https://books.google.com/books?id=E_wKgWBu8rUC&q=%22conductance&pg=PA105
 |isbn        = 978-0766820838
 |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
 |pages       = 60
 |url         = https://books.google.com/books?id=pRtTAAAAMAAJ&q=%22static+conductance%22+%22differential+conductance%22&pg=PA60
 |isbn        = 978-0123747006
 |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 (unit)|siemens]] (formerly ''mho'') which is the conductance of a resistor with a resistance of one [[ohm (unit)|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|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 ''(<span style="color:red;">red</span> region)''.<ref name="Simin" /> The differential resistance <math>r_\text{diff}</math> at a point '''''P''''' is the inverse slope of the line tangent to the graph at that point
<math>r_\text{diff} = \frac {\Delta v}{\Delta i} = \frac {v_2 - v_1}{i_2 - i_1}  </math><br />
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 ''(<span style="color:red;">red</span> region)'' current flows out of the positive terminal, so electric power flows out of the device into the circuit. For example at point '''''P''''', <math>v < 0</math> and <math>i > 0</math>, so<br />
<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        = 978-0521033336
 |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, <span style="color:red;">red</span>)''. It has negative differential resistance and negative static resistance (is active):<math display="block">R = \frac{\Delta v}{\Delta i} = \frac{v}{i} < 0</math>
| width4 = 187
}}