Electrical element

<h2 id="types">Types</h2>
Circuit elements can be classified into different categories. One is how many terminals they have to connect them to other components:

<ul><li>One-port elements – represent the simplest components, with only two terminals to connect to. Examples are
<ul><li><a href="/facts/Electrical_resistance_and_conductance/hxSt0UAj">resistances</a>,</li>
<li><a href="/facts/Capacitance/0pH046TL">capacitances</a>,</li>
<li><a href="/facts/Inductance/oIxj7JxX">inductances</a>,</li>
<li>and <a href="/facts/Diode/JyaoyGnk">diodes</a>.</li></ul></li>
<li>Two-port elements – are the most common multiport elements with four terminals consisting of two ports.</li>
<li>Multiport elements – these have more than two terminals. They connect to the external circuit through multiple pairs of terminals called <a href="/facts/Port_(circuit_theory)/g2mokqv0">ports</a>. For example,
<ul><li>a <a href="/facts/Transformer/EYYfXWrF">transformer</a> 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.</li></ul></li></ul>
Elements can also be divided into active and passive:

<ul><li>Passive elements – These elements do not have a source of energy; examples are
<ul><li>diodes,</li>
<li>resistances,</li>
<li>capacitances,</li>
<li>and inductances.</li></ul></li></ul>
<ul><li>Active elements or sources – these are elements which can source electrical <a href="/facts/Electric_power/w61ID9yQ">power</a>. They can be used to represent ideal <a href="/facts/Battery_(electricity)/bqEpm4ue">batteries</a> and <a href="/facts/Power_supply/4A8zBRHn">power supplies</a>; examples are
<ul><li><a href="/facts/Voltage_source/aAIbDymf">voltage sources</a></li>
<li>and <a href="/facts/Current_source/1UkYH0aG">current sources</a>.
<ul><li>Dependent sources – These are two-port elements with a voltage or current source proportional to the <a href="/facts/Voltage/EN2po6N2">voltage</a> or <a href="/facts/Electric_current/Dc1xdnUX">current</a> at a second pair of terminals. These are used in the modelling of <a href="/facts/Amplifier/pnRgxncC">amplifying</a> components such as
<ul><li><a href="/facts/Transistor/Bmr8zwL0">transistors</a>,</li>
<li><a href="/facts/Vacuum_tube/VHd1BNLq">vacuum tubes</a>,</li>
<li>and <a href="/facts/Op-amp/hunKdYb9">op-amps</a>.</li></ul></li></ul></li></ul></li></ul>
Another distinction is between linear and nonlinear:

<ul><li>Linear elements – these are elements in which the constituent relation, the relation between voltage and current, is a <a href="/facts/Linear_function/Ejvy8CKy">linear function</a>. They obey the <a href="/facts/Superposition_principle/GKGs9C92">superposition principle</a>. Examples of linear elements are resistances, capacitances, inductances, and linear-<a href="/facts/Dependent_source/dTx1CgCH">dependent sources</a>. <a href="/facts/Electrical_network/zt94kTW0">Circuits</a> with only linear elements, <a href="/facts/Linear_circuit/eDxQYWo2">linear circuits</a>, do not cause <a href="/facts/Intermodulation_distortion/R9ex1SMn">intermodulation distortion</a> and can be easily analysed with powerful mathematical techniques such as the <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a>.</li>
<li>Nonlinear elements – these are elements in which the relation between voltage and current is a <a href="/facts/Nonlinear_function/4aYItALC">nonlinear function</a>. An example is a <a href="/facts/Diode/JyaoyGnk">diode</a>, where the current is an <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> of the voltage. Circuits with nonlinear elements are harder to analyse and design, often requiring <a href="/facts/Circuit_simulation/HH9KUc0K">circuit simulation</a> computer programs such as <a href="/facts/SPICE/mIMfxvm7">SPICE</a>.</li></ul>
<h2 id="one-port-elements">One-port elements</h2>
Only nine types of element (<a href="/facts/Memristor/BrFKeE7l">memristor</a> not included), five passive and four active, are required to model any electrical component or circuit.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> Each element is defined by a relation between the <a href="/facts/State_variable/HNF1dCXu">state variables</a> of the network: <a href="/facts/Current_(electricity)/Dc1xdnUX">current</a>, 
 
 
 
 I
 
 
 {\displaystyle I}
 
; <a href="/facts/Voltage/EN2po6N2">voltage</a>, 
 
 
 
 V
 
 
 {\displaystyle V}
 
; <a href="/facts/Electric_charge/xxsBfmlB">charge</a>, 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
; and <a href="/facts/Magnetic_flux/jQMVxrnD">magnetic flux</a>, 
 
 
 
 Φ
 
 
 {\displaystyle \Phi }
 
.

<ul><li>Two sources:
<ul><li><a href="/facts/Current_source/1UkYH0aG">Current source</a>, measured in <a href="/facts/Ampere/rcKdfC2R">amperes</a> – produces a current in a conductor. Affects charge according to the relation 
 
 
 
 d
 Q
 =
 −
 I
 
 d
 t
 
 
 {\displaystyle dQ=-I\,dt}
 
.</li>
<li><a href="/facts/Voltage_source/aAIbDymf">Voltage source</a>, measured in <a href="/facts/Volt/KF18MXP2">volts</a> – produces a <a href="/facts/Potential_difference/EN2po6N2">potential difference</a> between two points. Affects magnetic flux according to the relation 
 
 
 
 d
 Φ
 =
 V
 
 d
 t
 
 
 {\displaystyle d\Phi =V\,dt}
 
.</li></ul></li></ul>

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 below.
<ul><li>Three <a href="/facts/Passivity_(engineering)/7oDeoqd3">passive</a> elements:
<ul><li><a href="/facts/Electrical_resistance/hxSt0UAj">Resistance</a> 
 
 
 
 R
 
 
 {\displaystyle R}
 
, measured in <a href="/facts/Ohm_(unit)/QNVrlGjw">ohms</a> – 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}
 
.</li>
<li><a href="/facts/Capacitance/0pH046TL">Capacitance</a> 
 
 
 
 C
 
 
 {\displaystyle C}
 
, measured in <a href="/facts/Farad/gPAYcVoE">farads</a> – 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}
 
.</li>
<li><a href="/facts/Inductance/oIxj7JxX">Inductance</a> 
 
 
 
 L
 
 
 {\displaystyle L}
 
, measured in <a href="/facts/Henry_(unit)/uwVv9zue">henries</a> – 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}
 
.</li></ul></li>
<li>Four abstract active elements:
<ul><li>Voltage-controlled voltage source (VCVS) Generates a voltage based on another voltage with respect to a specified gain. (has infinite input <a href="/facts/Electrical_impedance/I06NSXtN">impedance</a> and zero output impedance).</li>
<li>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 <a href="/facts/Field-effect_transistor/TerYC6Nz">field-effect transistors</a> and <a href="/facts/Vacuum_tube/VHd1BNLq">vacuum tubes</a> (has infinite input impedance and infinite output impedance). The gain is characterised by a <a href="/facts/Transfer_conductance/lwh8krQa">transfer conductance</a> which will have units of <a href="/facts/Siemens_(unit)/kLUHhcCj">siemens</a>.</li>
<li>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 <a href="/facts/Trancitor/sEyMuhbP">trancitors</a>. The gain is characterised by a <a href="/facts/Transfer_impedance/lwh8krQa">transfer impedance</a> which will have units of <a href="/facts/Ohm/QNVrlGjw">ohms</a>.</li>
<li>Current-controlled current source (CCCS) Generates a current based on an input current and a specified gain. Used to model <a href="/facts/Bipolar_junction_transistor/9BMcGv5Q">bipolar junction transistors</a>. (Has zero input impedance and infinite output impedance).</li></ul></li></ul>
These four elements are examples of two-port elements.
<h3>Non-linear elements</h3>

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 <a href="/facts/Constitutive_relation/IXTjEAbL">constitutive relation</a> 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 <a href="/facts/Memristor/BrFKeE7l">memristor</a>. 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 <a href="/facts/LTI_system_theory/kEmj0QDC">linear time-invariant (LTI)</a> circuit models. The constitutive relations of the passive elements are given by;<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<ul><li>Resistance: constitutive relation defined as 
 
 
 
 f
 (
 V
 ,
 I
 )
 =
 0
 
 
 {\displaystyle f(V,I)=0}
 
.</li>
<li>Capacitance: constitutive relation defined as 
 
 
 
 f
 (
 V
 ,
 Q
 )
 =
 0
 
 
 {\displaystyle f(V,Q)=0}
 
.</li>
<li>Inductance: constitutive relation defined as 
 
 
 
 f
 (
 Φ
 ,
 I
 )
 =
 0
 
 
 {\displaystyle f(\Phi ,I)=0}
 
.</li>
<li>Memristance: constitutive relation defined as 
 
 
 
 f
 (
 Φ
 ,
 Q
 )
 =
 0
 
 
 {\displaystyle f(\Phi ,Q)=0}
 
.</li></ul>
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 <a href="/facts/Diode/JyaoyGnk">diode</a>, 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.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
The fourth passive element, the memristor, was proposed by <a href="/facts/Leon_Chua/v5DWrV5E">Leon Chua</a> 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 <a href="/facts/HP_Labs/Pq3NGJh4">HP Labs</a> led by scientist <a href="/facts/R._Stanley_Williams/5ldcuuwv">R. Stanley Williams</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> 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:

<ul><li><a href="/facts/Nullator/RfNs2k1D">Nullator</a>: defined as 
 
 
 
 V
 =
 I
 =
 0
 
 
 {\displaystyle V=I=0}
 
</li>
<li><a href="/facts/Norator/zRGzUVz8">Norator</a>: defined as an element that places no restrictions on voltage and current whatsoever.</li></ul>
These are sometimes used in models of components with more than two terminals: transistors, for instance.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="two-port-elements">Two-port elements</h2>
All the above are two-terminal, or <a href="/facts/One-port/g2mokqv0">one-port</a>, elements except the dependent sources. Two lossless, passive, linear <a href="/facts/Two-port_network/cqUrBTYq">two-port</a> 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 <a href="/facts/Gyrator/YQFmuPbF">gyrator</a> 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 <a href="/facts/Reciprocity_(electrical_networks)/cFe3yYLT">reciprocal</a>. 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 <a href="/facts/Transformer/EYYfXWrF">transformer</a>, <a href="/facts/Inductor/YnfAXB3x">inductor</a>, and <a href="/facts/Capacitor/hyowqv7n">capacitor</a>, whereas a <a href="/facts/Gyrator/YQFmuPbF">practical gyrator</a> must be constructed as an active circuit.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="examples">Examples</h2>
The following are examples of representations of components by way of electrical elements.

<ul><li>On a first degree of approximation, a <a href="/facts/Battery_(electricity)/bqEpm4ue">battery</a> 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).</li>
<li>On a first degree of approximation, a <a href="/facts/Resistor/JrsGReM6">resistor</a> 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.</li>
<li>Current sources are often used when representing <a href="/facts/Semiconductor/zFf3mGTE">semiconductors</a>. For example, on a first degree of approximation, a bipolar <a href="/facts/Transistor/Bmr8zwL0">transistor</a> may be represented by a variable current source controlled by the input current.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Transmission_line/cYlqaPo3">Transmission line</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Thomas, Roland E.; Rosa, Albert J.; Toussaint, Gregory J. (2016). The Analysis and Design of Linear Circuits (8 ed.). Wiley. p. 17. ISBN 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. <a href="978-1-119-23538-5" target="_blank">978-1-119-23538-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Umesh, Rai (2007). "Bond graph toolbox for handling complex variable". IET Control Theory & Applications. 3 (5): 551–560. doi:10.1049/iet-cta.2007.0347. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Trajković, Ljiljana (2005). "Nonlinear circuits". In Chen, Wai-Kai (ed.). The Electrical Engineering Handbook. Academic Press. pp. 75–77. ISBN 0-12-170960-4. <a href="0-12-170960-4" target="_blank">0-12-170960-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Trajković, Ljiljana (2005). "Nonlinear circuits". In Chen, Wai-Kai (ed.). The Electrical Engineering Handbook. Academic Press. pp. 75–77. ISBN 0-12-170960-4. <a href="0-12-170960-4" target="_blank">0-12-170960-4</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Strukov, Dmitri B; Snider, Gregory S; Stewart, Duncan R; Williams, Stanley R (2008). "The missing memristor found". Nature. 453 (7191): 80–83. Bibcode:2008Natur.453...80S. doi:10.1038/nature06932. PMID 18451858. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">EETimes, 30 April 2008, 'Missing link' memristor created, EETimes, 30 April 2008 <a href="http://www.eetimes.com/news/latest/showArticle.jhtml?articleID=207403521" target="_blank">http://www.eetimes.com/news/latest/showArticle.jhtml?articleID=207403521</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Marks, Paul (30 April 2008). "Engineers find 'missing link' of electronics". New Scientist. <a href="https://www.newscientist.com/article/dn13812-engineers-find-missing-link-of-electronics.html" target="_blank">https://www.newscientist.com/article/dn13812-engineers-find-missing-link-of-electronics.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Researchers Prove Existence of New Basic Element for Electronic Circuits – 'Memristor' – 30 April 2008 <a href="http://www.physorg.com/news128786808.html" target="_blank">http://www.physorg.com/news128786808.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Trajković, Ljiljana (2005). "Nonlinear circuits". In Chen, Wai-Kai (ed.). The Electrical Engineering Handbook. Academic Press. pp. 75–77. ISBN 0-12-170960-4. <a href="0-12-170960-4" target="_blank">0-12-170960-4</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Wadhwa, C.L., Network analysis and synthesis, pp.17–22, New Age International, ISBN 81-224-1753-1. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Herbert J. Carlin, Pier Paolo Civalleri, Wideband circuit design, pp.171–172, CRC Press, 1998 ISBN 0-8493-7897-4. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">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. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
</ol>

Electrical element open-in-new

Electrical element