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Duality (electrical circuits)

Association of electrical terms into pairs based on interchanging voltage and current

Summary

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:

  • 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 (KCL) – Kirchhoff's voltage law (KVL)
  • 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.

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

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

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.
Wikipedia Source

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