duality (electrical circuits)

{{Short description|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.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}}.Alexander Russell, A Treatise on the Theory of Alternating Currents, volume 1, chapter XVII, Cambridge: University Press 1904 {{OCLC|264936988}}.

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.

Category:Electrical engineering

Electrical circuits