All about Duality

Duality

Two physical systems, or phenomenon are called dual if they are described by equations of the same mathematical form. The principle of duality holds goods for current and voltage, inductor and capacitor, series and parallel circuits. Some of the sets of duals are given below:

Element	Dual
Resistance	Conductance
Inductance	Capacitance
Series branch	Parallel branch
Voltage source	Current source
Loop	Node pair
Loop current	Node pair voltage
Mesh current	Node potential
Tie-set	Cut-set
Link	Twig
Short circuit	Open circuit
Series path	Parallel path
Switch closed(at t=0)	Switch opened (at t=0)

The two networks are said to be dual networks of each other if mesh (loop) equations of given network are node equations of the other. Dual networks are not equivalent networks of each other. Principal of duality only implies that circuit equations are of the same mathematical form and dual networks behave similarly but with interchanged variables.

Requirements For The Dual Networks

1. Number of meshes in a given network must be equal to the number of nodes of a dual network.
2. Total impedance of a given network must be equal to the total admittance of a dual network.
3. Impedance of a branch common to two meshes must be equal to the admittance between two nodes in the dual network.
4. Voltage source common to both the loops must be represented as current source between two nodes.
5. The switch opening at t=0 in the given network must be shown as a switch closing at t=0 in the dual network.