Duality (electrical circuits)

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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[edit]

The use of duality in circuit theory is due to Alexander Russell who published his ideas in 1904.[1][2]

Examples[edit]

Constitutive relations[edit]

  • 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[edit]

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[edit]

  • 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

See also[edit]

References[edit]

  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.