Editing Electrical element

{{short description|Idealized versions of real electronic components used in circuit analysis}}
{{Distinguish|Heating element}}
{{More references|date=August 2022}}

In [[electrical engineering]], '''electrical elements''' are conceptual abstractions representing idealized [[electrical component]]s,<ref name="ThomasRosaToussaint_2016">{{cite book | title = The Analysis and Design of Linear Circuits | edition = 8 | first1 = Roland E. | last1 = Thomas | first2 = Albert J. | last2 = Rosa | first3 = Gregory J. | last3 = Toussaint | publisher = Wiley | year = 2016 | page = 17 | isbn = 978-1-119-23538-5 | quote = To distinguish between a device (the real thing) and its model (an approximate stand-in), we call the model a circuit element. Thus, a device is an article of hardware described in manufacturers’ catalogs and parts specifications. An element is a model described in textbooks on circuit analysis.}}</ref> such as [[resistor]]s, [[capacitor]]s, and [[inductor]]s, used in [[circuit analysis|the analysis]] of [[electrical network]]s.  All electrical networks can be analyzed as multiple electrical elements interconnected by wires. Where the elements roughly correspond to real components, the representation can be in the form of a [[Schematic diagram#Electronic industry|schematic diagram]] or [[circuit diagram]]. This is called a [[lumped-element model|lumped-element circuit model]]. In other cases, infinitesimal elements are used to model the network in a [[distributed-element model]].

These ideal electrical elements represent actual, physical [[Electronic component|electrical or electronic components]]. Still, they do not exist physically and are assumed to have ideal properties. In contrast, actual electrical components have less than ideal properties, a degree of uncertainty in their values, and some degree of nonlinearity. To model the nonideal behavior of a real circuit component may require a combination of multiple ideal electrical elements to approximate its function. For example, an inductor circuit element is assumed to have [[inductance]] but no [[Electrical resistance and conductance|resistance]] or [[capacitance]], while a real inductor, a coil of wire, has some resistance in addition to its inductance. This may be modeled by an ideal inductance element in series with a resistance.

Circuit analysis using electric elements is useful for understanding practical networks of electrical components. Analyzing how a network is affected by its individual elements makes it possible to estimate how a real network will behave.

==Types==
Circuit elements can be classified into different categories. One is how many terminals they have to connect them to other components:
* '''''One-port elements'''''{{snd}} represent the simplest components, with only two terminals to connect to. Examples are 
** [[Electrical resistance and conductance|resistances]], 
** [[capacitance]]s, 
** [[inductance]]s, 
** and [[diode]]s.
* '''''Two-port elements'''''{{snd}} are the most common multiport elements with four terminals consisting of two ports.
* '''''Multiport elements'''''{{snd}}these have more than two terminals.  They connect to the external circuit through multiple pairs of terminals called [[port (circuit theory)|port]]s. For example, 
** a [[transformer]] with three separate windings has six terminals and could be idealized as a three-port element; the ends of each winding are connected to a pair of terminals representing a port.

Elements can also be divided into active and passive:

* '''''Passive elements'''''{{snd}}These elements do not have a source of energy; examples are 
** diodes, 
** resistances, 
** capacitances, 
** and inductances.

* '''''Active elements''''' or '''''sources'''''{{snd}}these are elements which can source electrical [[Electric power|power]]. They can be used to represent ideal [[battery (electricity)|batteries]] and [[power supply|power supplies]]; examples are 
** [[voltage source]]s 
** and [[current source]]s. 
*** '''''Dependent sources'''''{{snd}}These are two-port elements with a voltage or current source proportional to the [[voltage]] or [[Electric current|current]] at a second pair of terminals. These are used in the modelling of [[amplifier|amplifying]] components such as
**** [[transistor]]s, 
**** [[vacuum tube]]s, 
**** and [[op-amp]]s.

Another distinction is between linear and nonlinear:
* '''''Linear elements'''''{{snd}}these are elements in which the constituent relation, the relation between voltage and current, is a [[linear function]]. They obey the [[superposition principle]]. Examples of linear elements are resistances, capacitances, inductances, and linear-[[dependent source]]s. [[Electrical network|Circuits]] with only linear elements, [[linear circuit]]s, do not cause [[intermodulation distortion]] and can be easily analysed with powerful mathematical techniques such as the [[Laplace transform]].
* '''''Nonlinear elements'''''{{snd}}these are elements in which the relation between voltage and current is a [[nonlinear function]]. An example is a [[diode]], where the current is an [[exponential function]] of the voltage. Circuits with nonlinear elements are harder to analyse and design, often requiring [[circuit simulation]] computer programs such as [[SPICE]].

==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/>

==Two-port elements==
All the above are two-terminal, or [[one-port]], elements except the dependent sources. Two lossless, passive, linear [[two-port network|two-port]] elements are typically introduced into network analysis. Their constitutive relations in matrix notation are;

;Transformer:

: <math> \begin{bmatrix}  V_1  \\ I_2  \end{bmatrix} = \begin{bmatrix} 0 & n \\ -n & 0 \end{bmatrix}\begin{bmatrix} I_1  \\ V_2 \end{bmatrix}</math>

;Gyrator:

: <math> \begin{bmatrix}  V_1  \\ V_2  \end{bmatrix} = \begin{bmatrix} 0 & -r \\ r & 0 \end{bmatrix}\begin{bmatrix} I_1  \\ I_2 \end{bmatrix}</math>

The transformer maps a voltage at one port to a voltage at the other in a ratio of ''n''. The current between the same two ports is mapped by 1/''n''. On the other hand, the [[gyrator]] maps a voltage at one port to a current at the other. Likewise, currents are mapped to voltages. The quantity ''r'' in the matrix is in units of resistance. The gyrator is a necessary element in analysis because it is not [[Reciprocity (electrical networks)|reciprocal]]. Networks built from just the basic linear elements are necessarily reciprocal, so they cannot be used by themselves to represent a non-reciprocal system. It is not essential, however, to have both the transformer and gyrator. Two gyrators in cascade are equivalent to a transformer, but the transformer is usually retained for convenience. The introduction of the gyrator also makes either capacitance or inductance non-essential since a gyrator terminated with one of these at port 2 will be equivalent to the other at port 1. However, transformer, capacitance, and inductance are normally retained in analysis because they are the ideal properties of the basic physical components [[transformer]], [[inductor]], and [[capacitor]], whereas a [[Gyrator#Implementation: a simulated inductor|practical gyrator]] must be constructed as an active circuit.<ref>Wadhwa, C.L., ''Network analysis and synthesis'', pp.17–22, New Age International, {{ISBN|81-224-1753-1}}.</ref><ref>Herbert J. Carlin, Pier Paolo Civalleri, ''Wideband circuit design'', pp.171–172, CRC Press, 1998 {{ISBN|0-8493-7897-4}}.</ref><ref>Vjekoslav Damić, John Montgomery, ''Mechatronics by bond graphs: an object-oriented approach to modelling and simulation'', pp.32–33, Springer, 2003 {{ISBN|3-540-42375-3}}.</ref>

==Examples==
The following are examples of representations of components by way of electrical elements.
* On a first degree of approximation, a [[battery (electricity)|battery]] is represented by a voltage source. A more refined model also includes a resistance in series with the voltage source to represent the battery's internal resistance (which results in the battery heating and the voltage dropping when in use). A current source in parallel may be added to represent its leakage (which discharges the battery over a long period).
* On a first degree of approximation, a [[resistor]] is represented by a resistance. A more refined model also includes a series inductance to represent the effects of its lead inductance (resistors constructed as a spiral have more significant inductance). A capacitance in parallel may be added to represent the capacitive effect of the proximity of the resistor leads to each other. A wire can be represented as a low-value resistor.
* Current sources are often used when representing [[semiconductor]]s. For example, on a first degree of approximation, a bipolar [[transistor]] may be represented by a variable current source controlled by the input current.

==See also==
* [[Transmission line]]

==References==
{{Reflist}}

{{DEFAULTSORT:Electrical Element}}
[[Category:Electrical circuits]]
[[Category:Electrical systems]]

[[ar:عنصر كهربائي]]