# Duality (electrical circuits)

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)
${\displaystyle v=iR\iff i=vG\,}$
• Capacitor and inductor – differential form
${\displaystyle i_{C}=C{\frac {d}{dt}}v_{C}\iff v_{L}=L{\frac {d}{dt}}i_{L}}$
• Capacitor and inductor – integral form
${\displaystyle v_{C}(t)=V_{0}+{1 \over C}\int _{0}^{t}i_{C}(\tau )\,d\tau \iff i_{L}(t)=I_{0}+{1 \over L}\int _{0}^{t}v_{L}(\tau )\,d\tau }$

### Voltage division — current division

${\displaystyle v_{R_{1}}=v{\frac {R_{1}}{R_{1}+R_{2}}}\iff i_{G_{1}}=i{\frac {G_{1}}{G_{1}+G_{2}}}}$

• Resistor and conductor
${\displaystyle Z_{R}=R\iff Y_{G}=G}$
${\displaystyle Z_{G}={1 \over G}\iff Y_{R}={1 \over R}}$
• Capacitor and inductor
${\displaystyle Z_{C}={1 \over Cs}\iff Y_{L}={1 \over Ls}}$
${\displaystyle Z_{L}=Ls\iff Y_{c}=Cs}$