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) [math]\displaystyle{ v=iR \iff i=vG \, }[/math]
* Capacitor and inductor - differential form [math]\displaystyle{ i_C=C\frac{d}{dt}v_C \iff v_L=L\frac{d}{dt}i_L }[/math]
* Capacitor and inductor - integral form [math]\displaystyle{ 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 -- current division

[math]\displaystyle{ 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]\displaystyle{ Z_R = R \iff Y_G = G }[/math] [math]\displaystyle{ Z_G = {1 \over G } \iff Y_R = { 1 \over R } }[/math]
* Capacitor and inductor [math]\displaystyle{ Z_C = {1 \over Cs} \iff Y_L = {1 \over Ls} }[/math] [math]\displaystyle{ 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

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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