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 – serial (circuits)
• resistance – conductance
• 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

$Z_C = {1 \over Cs} \iff Y_L = {1 \over Ls}$

$Z_L = Ls \iff Y_c = Cs$

See also

• Duality (electricity and magnetism)
• Duality (mechanical engineering)
• Dual impedance
• Dual graph

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 XXI, Cambridge: University Press 1904 OCLC 264936988.

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