# Electrical element

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Idealized versions of real electronic components used in circuit analysis

Not to be confused with [Heating element](/source/Heating_element).

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In [electrical engineering](/source/Electrical_engineering), **electrical elements** are conceptual abstractions representing idealized [electrical components](/source/Electrical_component),[1] such as [resistors](/source/Resistor), [capacitors](/source/Capacitor), and [inductors](/source/Inductor), used in [the analysis](/source/Circuit_analysis) of [electrical networks](/source/Electrical_network). 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](/source/Schematic_diagram#Electronic_industry) or [circuit diagram](/source/Circuit_diagram). This is called a [lumped-element circuit model](/source/Lumped-element_model). In other cases, infinitesimal elements are used to model the network in a [distributed-element model](/source/Distributed-element_model).

These ideal electrical elements represent actual, physical [electrical or electronic components](/source/Electronic_component). 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](/source/Inductance) but no [resistance](/source/Electrical_resistance_and_conductance) or [capacitance](/source/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*** – represent the simplest components, with only two terminals to connect to. Examples are - [resistances](/source/Electrical_resistance_and_conductance), - [capacitances](/source/Capacitance), - [inductances](/source/Inductance), - and [diodes](/source/Diode).

- ***Two-port elements*** – are the most common multiport elements with four terminals consisting of two ports.

- ***Multiport elements*** – these have more than two terminals. They connect to the external circuit through multiple pairs of terminals called [ports](/source/Port_(circuit_theory)). For example, - a [transformer](/source/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*** – These elements do not have a source of energy; examples are - diodes, - resistances, - capacitances, - and inductances.

- ***Active elements*** or ***sources*** – these are elements which can source electrical [power](/source/Electric_power). They can be used to represent ideal [batteries](/source/Battery_(electricity)) and [power supplies](/source/Power_supply); examples are - [voltage sources](/source/Voltage_source) - and [current sources](/source/Current_source). - ***Dependent sources*** – These are two-port elements with a voltage or current source proportional to the [voltage](/source/Voltage) or [current](/source/Electric_current) at a second pair of terminals. These are used in the modelling of [amplifying](/source/Amplifier) components such as - [transistors](/source/Transistor), - [vacuum tubes](/source/Vacuum_tube), - and [op-amps](/source/Op-amp).

Another distinction is between linear and nonlinear:

- ***Linear elements*** – these are elements in which the constituent relation, the relation between voltage and current, is a [linear function](/source/Linear_function). They obey the [superposition principle](/source/Superposition_principle). Examples of linear elements are resistances, capacitances, inductances, and linear-[dependent sources](/source/Dependent_source). [Circuits](/source/Electrical_network) with only linear elements, [linear circuits](/source/Linear_circuit), do not cause [intermodulation distortion](/source/Intermodulation_distortion) and can be easily analysed with powerful mathematical techniques such as the [Laplace transform](/source/Laplace_transform).

This graph shows the nonlinearity of the current versus voltage curve of diodes.

- ***Nonlinear elements*** – these are elements in which the relation between voltage and current is a [nonlinear function](/source/Nonlinear_function). An example is a [diode](/source/Diode), where the current is an [exponential function](/source/Exponential_function) of the voltage. Circuits with nonlinear elements are harder to analyse and design, often requiring [circuit simulation](/source/Circuit_simulation) computer programs such as [SPICE](/source/SPICE).

## One-port elements

Only nine types of element ([memristor](/source/Memristor) not included), five passive and four active, are required to model any electrical component or circuit.[2] Each element is defined by a relation between the [state variables](/source/State_variable) of the network: [current](/source/Current_(electricity)), I {\displaystyle I} ; [voltage](/source/Voltage), V {\displaystyle V} ; [charge](/source/Electric_charge), Q {\displaystyle Q} ; and [magnetic flux](/source/Magnetic_flux), Φ {\displaystyle \Phi } .

- Two sources: - [Current source](/source/Current_source), measured in [amperes](/source/Ampere) – produces a current in a conductor. Affects charge according to the relation d Q = − I d t {\displaystyle dQ=-I\,dt} . - [Voltage source](/source/Voltage_source), measured in [volts](/source/Volt) – produces a [potential difference](/source/Potential_difference) between two points. Affects magnetic flux according to the relation d Φ = V d t {\displaystyle d\Phi =V\,dt} .

- - Φ {\displaystyle \Phi } in this relationship does not necessarily represent anything physically meaningful. In the case of the current generator, Q {\displaystyle Q} , the time integral of current represents the quantity of electric charge physically delivered by the generator. Here Φ {\displaystyle \Phi } 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](#Non-linear_elements) below.

- Three [passive](/source/Passivity_(engineering)) elements: - [Resistance](/source/Electrical_resistance) R {\displaystyle R} , measured in [ohms](/source/Ohm_(unit)) – produces a voltage proportional to the current flowing through the element. Relates voltage and current according to the relation d V = R d I {\displaystyle dV=R\,dI} . - [Capacitance](/source/Capacitance) C {\displaystyle C} , measured in [farads](/source/Farad) – produces a current proportional to the rate of change of voltage across the element. Relates charge and voltage according to the relation d Q = C d V {\displaystyle dQ=C\,dV} . - [Inductance](/source/Inductance) L {\displaystyle L} , measured in [henries](/source/Henry_(unit)) – produces the magnetic flux proportional to the rate of change of current through the element. Relates flux and current according to the relation d Φ = L d I {\displaystyle d\Phi =L\,dI} .

- 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 [impedance](/source/Electrical_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 transistors](/source/Field-effect_transistor) and [vacuum tubes](/source/Vacuum_tube) (has infinite input impedance and infinite output impedance). The gain is characterised by a [transfer conductance](/source/Transfer_conductance) which will have units of [siemens](/source/Siemens_(unit)). - 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 [trancitors](/source/Trancitor). The gain is characterised by a [transfer impedance](/source/Transfer_impedance) which will have units of [ohms](/source/Ohm). - Current-controlled current source (CCCS) Generates a current based on an input current and a specified gain. Used to model [bipolar junction transistors](/source/Bipolar_junction_transistor). (Has zero input impedance and infinite output impedance).

- - These four elements are examples of [two-port elements](#Two-port_elements).

### Non-linear elements

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](/source/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](/source/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 [linear time-invariant (LTI)](/source/LTI_system_theory) circuit models. The constitutive relations of the passive elements are given by;[3]

- Resistance: constitutive relation defined as f ( V , I ) = 0 {\displaystyle f(V,I)=0} .

- Capacitance: constitutive relation defined as f ( V , Q ) = 0 {\displaystyle f(V,Q)=0} .

- Inductance: constitutive relation defined as f ( Φ , I ) = 0 {\displaystyle f(\Phi ,I)=0} .

- Memristance: constitutive relation defined as f ( Φ , Q ) = 0 {\displaystyle f(\Phi ,Q)=0} .

- where f ( x , y ) {\displaystyle f(x,y)} 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](/source/Diode), which in circuit theory terms is a non-linear resistor, has a constitutive relation of the form V = f ( I ) {\displaystyle V=f(I)} . Both independent voltage and independent current sources can be considered non-linear resistors under this definition.[3]

The fourth passive element, the memristor, was proposed by [Leon Chua](/source/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](/source/HP_Labs) led by scientist [R. Stanley Williams](/source/R._Stanley_Williams).[4][5][6][7] 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](/source/Nullator): defined as V = I = 0 {\displaystyle V=I=0}

- [Norator](/source/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.[3]

## Two-port elements

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

**Transformer**

- [ V 1 I 2 ] = [ 0 n − n 0 ] [ I 1 V 2 ] {\displaystyle {\begin{bmatrix}V_{1}\\I_{2}\end{bmatrix}}={\begin{bmatrix}0&n\\-n&0\end{bmatrix}}{\begin{bmatrix}I_{1}\\V_{2}\end{bmatrix}}}

**Gyrator**

- [ V 1 V 2 ] = [ 0 − r r 0 ] [ I 1 I 2 ] {\displaystyle {\begin{bmatrix}V_{1}\\V_{2}\end{bmatrix}}={\begin{bmatrix}0&-r\\r&0\end{bmatrix}}{\begin{bmatrix}I_{1}\\I_{2}\end{bmatrix}}}

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](/source/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 [reciprocal](/source/Reciprocity_(electrical_networks)). 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](/source/Transformer), [inductor](/source/Inductor), and [capacitor](/source/Capacitor), whereas a [practical gyrator](/source/Gyrator#Implementation:_a_simulated_inductor) must be constructed as an active circuit.[8][9][10]

## Examples

The following are examples of representations of components by way of electrical elements.

- On a first degree of approximation, a [battery](/source/Battery_(electricity)) 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](/source/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 [semiconductors](/source/Semiconductor). For example, on a first degree of approximation, a bipolar [transistor](/source/Transistor) may be represented by a variable current source controlled by the input current.

## See also

- [Transmission line](/source/Transmission_line)

## References

1. **[^](#cite_ref-ThomasRosaToussaint_2016_1-0)** Thomas, Roland E.; Rosa, Albert J.; Toussaint, Gregory J. (2016). *The Analysis and Design of Linear Circuits* (8 ed.). Wiley. p. 17. [ISBN](/source/ISBN_(identifier)) [978-1-119-23538-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-119-23538-5). 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.

1. **[^](#cite_ref-Umesh_2-0)** Umesh, Rai (2007). "Bond graph toolbox for handling complex variable". *IET Control Theory & Applications*. **3** (5): 551–560. [doi](/source/Doi_(identifier)):[10.1049/iet-cta.2007.0347](https://doi.org/10.1049%2Fiet-cta.2007.0347).

1. ^ [***a***](#cite_ref-Trajkovic_3-0) [***b***](#cite_ref-Trajkovic_3-1) [***c***](#cite_ref-Trajkovic_3-2) Trajković, Ljiljana (2005). "Nonlinear circuits". In Chen, Wai-Kai (ed.). *The Electrical Engineering Handbook*. Academic Press. pp. 75–77. [ISBN](/source/ISBN_(identifier)) [0-12-170960-4](https://en.wikipedia.org/wiki/Special:BookSources/0-12-170960-4).

1. **[^](#cite_ref-4)** Strukov, Dmitri B; Snider, Gregory S; Stewart, Duncan R; Williams, Stanley R (2008). "The missing memristor found". *Nature*. **453** (7191): 80–83. [Bibcode](/source/Bibcode_(identifier)):[2008Natur.453...80S](https://ui.adsabs.harvard.edu/abs/2008Natur.453...80S). [doi](/source/Doi_(identifier)):[10.1038/nature06932](https://doi.org/10.1038%2Fnature06932). [PMID](/source/PMID_(identifier)) [18451858](https://pubmed.ncbi.nlm.nih.gov/18451858).

1. **[^](#cite_ref-5)** EETimes, 30 April 2008, ['Missing link' memristor created](http://www.eetimes.com/news/latest/showArticle.jhtml?articleID=207403521), EETimes, 30 April 2008

1. **[^](#cite_ref-6)** Marks, Paul (30 April 2008). ["Engineers find 'missing link' of electronics"](https://www.newscientist.com/article/dn13812-engineers-find-missing-link-of-electronics.html). *New Scientist*.

1. **[^](#cite_ref-7)** [Researchers Prove Existence of New Basic Element for Electronic Circuits – 'Memristor'](http://www.physorg.com/news128786808.html) – 30 April 2008

1. **[^](#cite_ref-8)** Wadhwa, C.L., *Network analysis and synthesis*, pp.17–22, New Age International, [ISBN](/source/ISBN_(identifier)) [81-224-1753-1](https://en.wikipedia.org/wiki/Special:BookSources/81-224-1753-1).

1. **[^](#cite_ref-9)** Herbert J. Carlin, Pier Paolo Civalleri, *Wideband circuit design*, pp.171–172, CRC Press, 1998 [ISBN](/source/ISBN_(identifier)) [0-8493-7897-4](https://en.wikipedia.org/wiki/Special:BookSources/0-8493-7897-4).

1. **[^](#cite_ref-10)** Vjekoslav Damić, John Montgomery, *Mechatronics by bond graphs: an object-oriented approach to modelling and simulation*, pp.32–33, Springer, 2003 [ISBN](/source/ISBN_(identifier)) [3-540-42375-3](https://en.wikipedia.org/wiki/Special:BookSources/3-540-42375-3).

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Adapted from the Wikipedia article [Electrical element](https://en.wikipedia.org/wiki/Electrical_element) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Electrical_element?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.