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| '''Electrical elements''' are conceptual abstractions representing [[idealized]] [[electrical component]]s, such as [[resistor]]s, [[capacitor]]s, and [[inductor]]s, used in the [[circuit analysis|analysis]] of [[electrical network]]s. Any electrical network can be analysed as multiple, interconnected electrical elements in a [[Schematic diagram#Electronic industry|schematic diagram]] or [[circuit diagram]], each of which affects the [[voltage]] in the network or [[Current (electricity)|current]] through the network. These ideal electrical elements represent real, physical [[Electronic component|electrical or electronic components]] but they do not exist physically and they are assumed to have ideal properties according to a [[lumped element model]], while components are objects with less than ideal properties, a degree of uncertainty in their values and some degree of nonlinearity, each of which may require a combination of multiple electrical elements in order to approximate its function.
| | The one who wrote the article is called Lida. Releasing output is her day-job currently and her pay has been definitely fulfilling. Her husband and her livein West Virginia. Kayaking is what she loves doing.<br><br>My web page: [http://www.linkedin.com/pub/jordan-kurland/a/618/581 Jordan Kurland] |
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| Circuit analysis using electric elements is useful for understanding many practical electrical networks using components. By analyzing the way a network is affected by its individual elements it is possible to estimate how a real network will behave.
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| ==One-port elements==
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| Only nine types of element ([[memristor]] not included), five passive and four active, are required to model any electrical component or circuit.{{Citation needed|date=March 2012}} 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>.
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| * Two sources:
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| **[[Current source]], measured in [[ampere]]s – produces a current in a conductor. Affects charge according to the relation <math>dQ = -I\,dt</math>.
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| **[[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>.
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| ::<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.
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| ::Both these elements are necessarily non-linear elements. See [[#Non-linear elements]] below.
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| * Three [[Passivity (engineering)|passive]] elements:
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| **[[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>.
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| **[[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>.
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| **[[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>.
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| * Four abstract active elements:
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| ** Voltage-controlled voltage source (VCVS) Generates a voltage based on another voltage with respect to a specified gain. (has infinite input impedance and zero output impedance).
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| ** 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]].
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| ** 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). The gain is characterised by a [[transfer impedance]] which will have units of [[ohm]]s.
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| ** 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).
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| ::These four elements are examples of [[#Two-port elements|two-port elements]].
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| ===Non-linear elements===
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| In reality, all circuit components are non-linear and can only be approximated to linear over a certain range. To more exactly describe the passive elements, their [[constitutive relation]] is used instead of simple proportionality. From any two of the circuit variables there are six constitutive relations that can be formed. From this it is supposed that there is 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. 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>
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| *Resistance: constitutive relation defined as <math>f(V, I)=0</math>.
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| *Capacitance: constitutive relation defined as <math>f(V, Q)=0</math>.
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| *Inductance: constitutive relation defined as <math>f(\Phi, I)=0</math>.
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| *Memristance: constitutive relation defined as <math>f(\Phi, Q)=0</math>.
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| :where <math>f(x,y)</math> is an arbitrary function of two variables.
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| 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/>
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| 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|last=Strukov|first=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|url=http://www.nature.com/nature/journal/v453/n7191/full/nature06932.html|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>[http://technology.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. Because memristors are time-variant by definition, they are not included in [[LTI system theory|linear time-invariant (LTI)]] circuit models.{{cn|reason=they are not included because they are not linear surely. It is perfectly feasible to have a time invariant memristor as a theoretical ideal element|date=May 2013}}
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| There are also two special non-linear elements which are sometimes used in analysis but which are not the ideal counterpart of any real component:
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| *[[Nullator]]: defined as <math> V = I = 0 </math>
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| *[[Norator]]: defined as an element which places no restrictions on voltage and current whatsoever.
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| These are sometimes used in models of components with more than two terminals: transistors for instance.<ref name=Trajkovic/>
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| ==Two-port elements==
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| All the above are two-terminal, or one-port, elements with the exception of the dependent sources. There are two lossless, passive, linear [[two-port network|two-port]] elements that are normally introduced into network analysis. Their constitutive relations in matrix notation are;
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| ;Transformer:
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| : <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>
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| ;Gyrator:
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| : <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>
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| 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 port is mapped by 1/''n''. The [[gyrator]], on the other hand, 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_(electromagnetism)#Reciprocity_for_electrical_networks|reciprocal]]. Networks built from the basic linear elements only are obliged to be reciprocal and so 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. 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>
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| ==Examples==
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| The following are examples of representation of components by way of electrical elements.
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| * 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 of time).
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| * 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
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| * Current sources are more 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 that is controlled by the input current.
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| ==See also==
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| {{Commons|Electronic devices}}
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| * [[Electronic component]]
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| * [[Lumped element model]]
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| * [[Distributed element model]]
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| * [[Transmission line]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Electrical Element}}
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| [[Category:Electronics]]
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| [[Category:Electrical systems]]
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| [[ar:عنصر كهربائي]]
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| [[ta:மின் உறுப்புகள்]]
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The one who wrote the article is called Lida. Releasing output is her day-job currently and her pay has been definitely fulfilling. Her husband and her livein West Virginia. Kayaking is what she loves doing.
My web page: Jordan Kurland