Duality (electrical circuits)

Short description: Association of electrical terms into pairs based on interchanging voltage and current

In electrical engineering, electrical terms are associated into pairs called duals. A dual of a relationship is formed by interchanging voltage and 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 of electricity and magnetism.

Here is a partial list of electrical dualities:

* voltage - current
* parallel - series (circuits)
* resistance - conductance
* voltage division - current division
* impedance - admittance
* capacitance - inductance
* reactance - susceptance
* short circuit - open circuit
* Kirchhoff's current law - Kirchhoff's voltage law.
* Thévenin's theorem - Norton's theorem

## Contents

* 1 History
* 2 Examples
* 2.1 Constitutive relations
* 2.2 Voltage division — current division
* 2.3 Impedance and admittance
* 3 See also
* 4 References

## History

The use of duality in circuit theory is due to Alexander Russell who published his ideas in 1904.[1][2]

## Examples

### Constitutive relations

* Resistor and conductor (Ohm's law) v=iR⟺i=vG
* Capacitor and inductor - differential form iC=CddtvC⟺vL=LddtiL
* Capacitor and inductor - integral form vC(t)=V0+1C∫0tiC(τ)dτ⟺iL(t)=I0+1L∫0tvL(τ)dτ

### Voltage division -- current division

vR1=vR1R1+R2⟺iG1=iG1G1+G2

### Impedance and admittance

* Resistor and conductor ZR=R⟺YG=G ZG=1G⟺YR=1R
* Capacitor and inductor ZC=1Cs⟺YL=1Ls ZL=Ls⟺Yc=Cs

## See also

* Duality (electricity and magnetism)
* Duality (mechanical engineering)
* Dual impedance
* Dual graph
* Mechanical–electrical analogies
* List of dualities

## References

1. ↑ Belevitch, V, "Summary of the history of circuit theory", Proceedings of the IRE, vol 50, Iss 5, pp. 848–855, May 1962 doi:10.1109/JRPROC.1962.288301.
2. ↑ Alexander Russell, A Treatise on the Theory of Alternating Currents, volume 1, chapter XXI, Cambridge: University Press 1904 OCLC 264936988.

* Turner, Rufus P, Transistors Theory and Practice, Gernsback Library, Inc, New York, 1954, Chapter 6.

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