Duality (electrical circuits) explained

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. KVL and KCL
• 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) $$v=iR\; \backslash iff\; i=vG\; \backslash ,$$
• Capacitor and inductor  - differential form $$i\_C=C\backslash fracv\_C\; \backslash iff\; v\_L=L\backslash fraci\_L$$
• Capacitor and inductor  - integral form $$v\_C(t)\; =\; V\_0\; +\; \backslash int\_^\; i\_C(\backslash tau)\; \backslash ,\; d\backslash tau\; \backslash iff\; i\_L(t)\; =\; I\_0\; +\; \backslash int\_^\; v\_L(\backslash tau)\; \backslash ,\; d\backslash tau$$

Voltage division - current division

$$v\_=v\backslash frac\; \backslash iff\; i\_=i\backslash frac$$

Impedance and admittance

• Resistor and conductor $$Z\_R\; =\; R\; \backslash iff\; Y\_G\; =\; G$$ $$Z\_G\; =\; \backslash iff\; Y\_R\; =$$
• Capacitor and inductor $$Z\_C\; =\; \backslash iff\; Y\_L\; =$$ $$Z\_L\; =\; Ls\; \backslash iff\; Y\_c\; =\; Cs$$

See also

• Duality (electricity and magnetism)
• Duality (mechanical engineering)
• Dual impedance
• Dual graph
• Mechanical–electrical analogies
• List of dualities

References

• Turner, Rufus P, Transistors Theory and Practice, Gernsback Library, Inc, New York, 1954, Chapter 6.

Notes and References

1. Belevitch, V, "Summary of the history of circuit theory", Proceedings of the IRE, vol 50, Iss 5, pp. 848–855, May 1962 .
2. Alexander Russell, A Treatise on the Theory of Alternating Currents, volume 1, chapter XVII, Cambridge: University Press 1904 .