# Duality (electrical circuits)

(Redirected from Duality (electrical engineering))

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.

## 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
$Z_{C}={1 \over Cs}\iff Y_{L}={1 \over Ls}$
$Z_{L}=Ls\iff Y_{c}=Cs$