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:

• voltage – current
• parallel – serial (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

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

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

Impedance and admittance

• 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

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

${\displaystyle Z_{L}=Ls\iff Y_{c}=Cs}$

See also

• Duality (electricity and magnetism)
• Duality (mechanical engineering)
• Dual impedance
• Dual graph
• 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.