Editing Electrical element (section)

==One-port elements==
Only nine types of element ([[memristor]] not included), five passive and four active, are required to model any electrical component or circuit.<ref name="Umesh">{{cite journal |last1=Umesh |first1=Rai |title=Bond graph toolbox for handling complex variable |journal=IET Control Theory & Applications |date=2007 |volume=3 |issue=5 |pages=551–560 |doi=10.1049/iet-cta.2007.0347}}</ref>  Each element is defined by a relation between the [[state variable]]s of the network: [[Current (electricity)|current]], <math>I</math>; [[voltage]], <math>V</math>; [[Electric charge|charge]], <math>Q</math>; and [[magnetic flux]], <math>\Phi</math>.
* Two sources:
** [[Current source]], measured in [[ampere]]s – produces a current in a conductor.  Affects charge according to the relation <math>dQ = -I\,dt</math>.
** [[Voltage source]], measured in [[volt]]s – produces a [[potential difference]] between two points.  Affects magnetic flux according to the relation <math>d\Phi = V\,dt</math>.
::<math>\Phi</math> in this relationship does not necessarily represent anything physically meaningful. In the case of the current generator, <math>Q</math>, the time integral of current represents the quantity of electric charge physically delivered by the generator. Here <math>\Phi</math> is the time integral of voltage, but whether or not that represents a physical quantity depends on the nature of the voltage source. For a voltage generated by magnetic induction, it is meaningful, but for an electrochemical source, or a voltage that is the output of another circuit, no physical meaning is attached to it.
::Both these elements are necessarily non-linear elements. See [[#Non-linear elements]] below.
* Three [[Passivity (engineering)|passive]] elements:
** [[Electrical resistance|Resistance]] <math>R</math>, measured in [[Ohm (unit)|ohms]] – produces a voltage proportional to the current flowing through the element. Relates voltage and current according to the relation <math>dV = R\,dI</math>.
** [[Capacitance]] <math>C</math>, measured in [[farad]]s – produces a current proportional to the rate of change of voltage across the element. Relates charge and voltage according to the relation <math>dQ = C\,dV</math>.
** [[Inductance]] <math>L</math>, measured in [[Henry (unit)|henries]] – produces the magnetic flux proportional to the rate of change of current through the element. Relates flux and current according to the relation <math>d\Phi = L\,dI</math>.
* Four abstract active elements:
** Voltage-controlled voltage source (VCVS) Generates a voltage based on another voltage with respect to a specified gain. (has infinite input [[Electrical impedance|impedance]] and zero output impedance).
** Voltage-controlled current source (VCCS) Generates a current based on a voltage elsewhere in the circuit, with respect to a specified gain, used to model [[field-effect transistor]]s and [[vacuum tube]]s (has infinite input impedance and infinite output impedance). The gain is characterised by a [[transfer conductance]] which will have units of [[Siemens (unit)|siemens]].
** Current-controlled voltage source (CCVS) Generates a voltage based on an input current elsewhere in the circuit with respect to a specified gain. (has zero input impedance and zero output impedance). Used to model [[trancitor]]s.  The gain is characterised by a [[transfer impedance]] which will have units of [[ohm]]s.
** Current-controlled current source (CCCS) Generates a current based on an input current and a specified gain. Used to model [[bipolar junction transistor]]s. (Has zero input impedance and infinite output impedance).
::These four elements are examples of [[#Two-port elements|two-port elements]].

===Non-linear elements===
[[File:Two-terminal non-linear circuit elements.svg|thumb|right|Conceptual symmetries of resistor, capacitor, inductor, and memristor.]]
In reality, all circuit components are non-linear and can only be approximated as linear over a certain range. To describe the passive elements more precisely, their [[constitutive relation]] is used instead of simple proportionality. Six constitutive relations can be formed from any two of the circuit variables. From this, there is supposed to be a theoretical fourth passive element since there are only five elements in total (not including the various dependent sources) found in linear network analysis. This additional element is called [[memristor]]. It only has any meaning as a time-dependent non-linear element; as a time-independent linear element, it reduces to a regular resistor. Hence, it is not included in [[LTI system theory|linear time-invariant (LTI)]] circuit models. The constitutive relations of the passive elements are given by;<ref name=Trajkovic>Ljiljana Trajković, "Nonlinear circuits", ''The Electrical Engineering Handbook'' (Ed: Wai-Kai Chen), pp.75–77, Academic Press, 2005 {{ISBN|0-12-170960-4}}</ref>
* Resistance: constitutive relation defined as <math>f(V, I)=0</math>.
* Capacitance: constitutive relation defined as <math>f(V, Q)=0</math>.
* Inductance: constitutive relation defined as <math>f(\Phi, I)=0</math>.
* Memristance: constitutive relation defined as <math>f(\Phi, Q)=0</math>.

:where <math>f(x,y)</math> is an arbitrary function of two variables.

In some special cases, the constitutive relation simplifies to a function of one variable. This is the case for all linear elements, but also, for example, an ideal [[diode]], which in circuit theory terms is a non-linear resistor, has a constitutive relation of the form <math> V = f(I)</math>. Both independent voltage and independent current sources can be considered non-linear resistors under this definition.<ref name=Trajkovic/>

The fourth passive element, the memristor, was proposed by [[Leon Chua]] in a 1971 paper, but a physical component demonstrating memristance was not created until thirty-seven years later. It was reported on April 30, 2008, that a working memristor had been developed by a team at [[HP Labs]] led by scientist [[R. Stanley Williams]].<ref>{{citation|last1=Strukov|first1=Dmitri B|last2=Snider|first2=Gregory S|last3=Stewart|first3=Duncan R|last4=Williams|first4=Stanley R|title=The missing memristor found|journal=Nature|volume=453|pages=80–83|year=2008|doi=10.1038/nature06932|pmid=18451858|issue=7191|bibcode=2008Natur.453...80S}}</ref><ref>EETimes, 30 April 2008, [http://www.eetimes.com/news/latest/showArticle.jhtml?articleID=207403521 'Missing link' memristor created], EETimes, 30 April 2008</ref><ref>[https://www.newscientist.com/article/dn13812-engineers-find-missing-link-of-electronics.html Engineers find 'missing link' of electronics] – 30 April 2008</ref><ref>[http://www.physorg.com/news128786808.html Researchers Prove Existence of New Basic Element for Electronic Circuits – 'Memristor'] – 30 April 2008</ref>  With the advent of the memristor, each pairing of the four variables can now be related.

Two special non-linear elements are sometimes used in analysis but are not the ideal counterpart of any real component:
* [[Nullator]]: defined as <math> V = I  = 0 </math>
* [[Norator]]: defined as an element that places no restrictions on voltage and current whatsoever.

These are sometimes used in models of components with more than two terminals: transistors, for instance.<ref name=Trajkovic/>