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Duality (electrical circuits)

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(Redirected from Duality (electrical engineering)) Association of electrical terms into pairs based on interchanging voltage and current

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 = i R ⟺ i = v G {\displaystyle v=iR\iff i=vG\,}
  • Capacitor and inductor – differential form i C = C d d t v C ⟺ v L = L d d t i L {\displaystyle 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 C ∫ 0 t i C ( τ ) d τ ⟺ i L ( t ) = I 0 + 1 L ∫ 0 t v L ( τ ) d τ {\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

v R 1 = v R 1 R 1 + R 2 ⟺ i G 1 = i G 1 G 1 + G 2 {\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 Z R = R ⟺ Y G = G {\displaystyle Z_{R}=R\iff Y_{G}=G} Z G = 1 G ⟺ Y R = 1 R {\displaystyle Z_{G}={1 \over G}\iff Y_{R}={1 \over R}}
  • Capacitor and inductor Z C = 1 C s ⟺ Y L = 1 L s {\displaystyle Z_{C}={1 \over Cs}\iff Y_{L}={1 \over Ls}} Z L = L s ⟺ Y c = C s {\displaystyle Z_{L}=Ls\iff Y_{c}=Cs}

See also

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 XVII, 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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