View source for Duality (electrical circuits)

{{Short description|Association of electrical terms into pairs based on interchanging voltage and current}}
{{Use dmy dates|date=January 2020|cs1-dates=y}}

In [[electrical engineering]], electrical terms are associated into pairs called '''duals'''.  A dual of a relationship is formed by interchanging [[voltage]] and [[Electric current|current]] in an expression. The dual expression thus produced is of the same form, and the reason that the dual is always a valid statement can be traced to the [[Duality (electricity and magnetism)|duality of electricity and magnetism]].

Here is a partial list of electrical dualities:

* voltage &ndash; current
* [[Series and parallel circuits|parallel &ndash; series]] (circuits)
* [[Electrical resistance|resistance]] &ndash; [[Electrical conductance|conductance]]
* [[voltage divider|voltage division]] &ndash; [[current divider|current division]]
* [[Electrical impedance|impedance]] &ndash; [[admittance]]
* [[capacitance]] &ndash; [[inductance]]
* [[Reactance (electronics)|reactance]] &ndash; [[susceptance]]
* [[short circuit]] &ndash; [[wikt:open circuit|open circuit]]
* [[Kirchhoff's current law]] &ndash; [[Kirchhoff's voltage law]].
* [[Thévenin's theorem]] &ndash; [[Norton's theorem]]

== History ==
The use of duality in [[circuit theory]] is due to Alexander Russell who published his ideas in 1904.<ref>Belevitch, V, "Summary of the history of circuit theory", ''Proceedings of the IRE'', '''vol 50''', Iss 5, pp.&nbsp;848–855, May 1962 {{doi|10.1109/JRPROC.1962.288301}}.</ref><ref>Alexander Russell, ''A Treatise on the Theory of Alternating Currents'', volume 1, chapter XXI, Cambridge: University Press 1904 {{OCLC|264936988}}.</ref>

== Examples ==
=== Constitutive relations ===

* Resistor and conductor (Ohm's law) <math display="block">v=iR \iff i=vG \, </math>
* Capacitor and inductor&nbsp;&ndash; differential form <math display="block">i_C=C\frac{d}{dt}v_C \iff v_L=L\frac{d}{dt}i_L</math>
* Capacitor and inductor&nbsp;&ndash; integral form <math display="block">v_C(t) = V_0 + {1 \over C}\int_{0}^{t} i_C(\tau) \, d\tau \iff  i_L(t) = I_0 + {1 \over L}\int_{0}^{t} v_L(\tau) \, d\tau </math>

=== Voltage division &mdash; current division ===

<math display="block">v_{R_1}=v\frac{R_1}{R_1 + R_2} \iff i_{G_1}=i\frac{G_1}{G_1 + G_2}</math>

=== Impedance and admittance ===

* Resistor and conductor <math display="block">Z_R = R \iff Y_G = G </math> <math display="block">Z_G = {1 \over G } \iff Y_R = { 1 \over R } </math>
* Capacitor and inductor <math display="block"> Z_C = {1 \over Cs} \iff  Y_L = {1 \over Ls} </math> <math display="block"> Z_L = Ls  \iff  Y_c = Cs</math>

== See also ==
* [[Duality (electricity and magnetism)]]
* [[Duality (mechanical engineering)]]
* [[Dual impedance]]
* [[Dual graph]]
* [[Mechanical–electrical analogies]]
* [[List of dualities]]

== References ==
{{reflist}}
* Turner, Rufus P, ''Transistors Theory and Practice'', Gernsback Library, Inc, New York, 1954, Chapter 6.

[[Category:Electrical engineering]]
[[Category:Duality theories|Electrical circuits]]