# Duality (electrical circuits)

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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:

## 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)
$v=iR \iff i=vG \,$
• Capacitor and inductor – differential form
$i_C=C\frac{d}{dt}v_C \iff v_L=L\frac{d}{dt}i_L$
• Capacitor and inductor – integral form
$v_C(t) = V_0 + {1 \over C}\int_{0}^{t} i_C(\tau) \, d\tau \iff i_L(t) = I_0 + {1 \over L}\int_{0}^{t} v_L(\tau) \, d\tau$

### Voltage division — current division

$v_{R_1}=v\frac{R_1}{R_1 + R_2} \iff i_{G_1}=i\frac{G_1}{G_1 + G_2}$

### Impedance and admittance

• Resistor and conductor
$Z_R = R \iff Y_G = G$
$Z_G = {1 \over G } \iff Y_R = { 1 \over R }$
• Capacitor and inductor
$Z_C = {1 \over Cs} \iff Y_L = {1 \over Ls}$
$Z_L = Ls \iff Y_c = Cs$

## 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.