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 – series (circuits)
 * resistance – conductance
 * voltage division – current division
 * impedance – admittance
 * capacitance – inductance
 * reactance – susceptance
 * short circuit – open circuit
 * Kirchhoff's current law – Kirchhoff's voltage law. KVL and KCL
 * 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.

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 &mdash; 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 $$ Z_C = {1 \over Cs} \iff Y_L = {1 \over Ls} $$ $$ Z_L = Ls  \iff  Y_c = Cs$$