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 – 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

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

↑ 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.
↑ 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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[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.

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